Sunex: In the original canon, this is all handled by an entity called CenCom, which is sort of ambiguously governmental; it's presented as having the types and degrees of control over the lives of the people who work for it that you'd expect from a branch of the military, and some trappings of that - uniforms, hierarchy, psych profiles, the approach to task assignments - but it's never actually specified, and in other ways it's treated more like a regular company. (The fact that the first of the stories was written in the early '60s may be relevant.) I've replaced CenCom outright with Sunex, which is straightforwardly a company and does many other things besides raise and educate shellpeople; it's inspired by Google for scope and Apple for attitude. In particular Sunex is very focused on brand loyalty, to the point of pulling the same sort of shenanigans Apple does, where you have to get a Sunex brand charger for your Sunex brand gadget and can't just use a regular USB charger, and to get your gadget repaired you have to go to a Sunex brand repair place and pay heavily for the privilege. They have the Apple-like focus on user experience, too, but your average shellperson would find this much less appealing of a tradeoff if they hadn't been raised from infancy to like the Sunex brand - and also of course the shells, ships, buildings, and everything else that they're installed in is all Sunex tech and requires Sunex brand upkeep no matter what they happen to think about it.
Financials and careers: Every newly graduated shellperson starts out in debt to Sunex, for their medical care and education and sometimes whatever thing they've ultimately decided to be; this works out to the equivalent of a few million dollars on the low end, and quite a bit more for ships (a commercial jet costs $50 million; that may be a little high - brainships are smaller - but it's in the right ballpark, I think, for a spaceship in a culture where those are as common as jets) and the rare individually-owned space station. Most shellpeople opt to take groundside jobs; the ratio is something like 50% building, 20% abstract, 20% space station, 5% city, 5% ship. (City jobs are prestigious and there's competition for them; abstract jobs vary significantly but tend to be lowest prestige, though some building jobs are also pretty low. Being a spaceship is glamorous, but not automatically prestigious.) Most jobs are straightforwardly salaried, and Sunex simply gets a cut of the shellperson's pay until their debt is paid off; for the jobs that aren't, it works more like a mortgage, where there's a certain amount that needs to be paid every so often (with accommodation for the fact that ships' jobs often have them out of contact with the rest of the universe for months at a time) and if you get too far behind they repossess the thing - except that in this case the thing in question is the shellperson's, uh, body, so what actually happens is that they become a de facto Sunex employee, doing whatever tasks they're assigned until they're not behind any more - and Sunex likes when this happens, so things are set up so that it's relatively easy to fall into that trap if you're careless or unlucky.
Brawns: Most shellpeople have an ablebodied partner, known as a brawn. Brawns' responsibilities vary depending on the type of work their shellperson partner does, but in general anything that needs hands, mobility on a human scale, face to face interaction, or a second opinion or second set of eyes will be the brawn's responsibility. Brawns' training varies depending on which type of shellperson they intend to eventually work with. In the original canon brawns were CenCom employees, assigned to ships on the basis of psych profiles; in my version, brawns are almost universally trained by Sunex, but this is more like getting a degree than like completing basic training for the army; once they have the qualification, getting hired is a separate thing that doesn't directly involve Sunex. For shellpeople hired by a company or the government (as most of them are), the company will generally also hire a brawn, with the shellperson being highly involved in the process and having significant veto power; independent shellpeople (most ships; occasionally ones with abstract jobs; very rarely stations) can either straightforwardly hire a brawn as an employee, or take one on as a business partner: the latter is safer financially, as the brawn traditionally gets half of whatever income they have after operating expenses no matter how little that may be, but it also gives the brawn quite a bit more input into what jobs they take, how they do things, etc.
Tech: The tech level is a bit lower than Star Trek but generally similar. There aren't teleporters; there are food-only replicators, but they need specialized stock. Nonperson AIs exist; artificial gravity does, too. FTL exists and both brainships and AI-controled ships can travel that way; singularity jumping (teleporting from a singularity-favorable spot to its counterpart elsewhere, similar to Vorkosigan-verse wormholes) is newer and only specially equipped brainships can do it. (Singularity ships are very expensive.) Brawns are commonly equipped with an implant that lets their partner look through their eyes and speak privately to them. Various robots (called servos) exist for shellpeople to use, but are relatively limited in what they can do, particularly in terms of strength and ability to handle difficult terrain. Holographic 3d video ('tri-vid') is common. Portable data storage takes the form of 'hedrons' which are variously sized depending on how much data they can hold (minihedrons are small enough to be be incorporated into jewelry; megahedrons have been mentioned) and are described as shiny black. Prosthetic limbs and organs exist, including ones with extra abilities that the original part might not have, for example a liver that can be set to filter a particular drug out of the bloodstream especially quickly; these are expensive and offer less sensory feedback than the real thing. People who have access to the best medical care can expect to live to 180 or 200; in the original canon ships' brawns are expected to retire at 75 and this is implied to be quite early. (Shellpeople don't die of old age and are very hard to damage.)
Space! It has aliens in: It does! None of them are fleshed out at this point. Some of them are humanoid, some are very much not; some are higher tech than humans were when the humans found them, some are very much not. I'm keeping the thing from the original canon where humans bought the energy tech that allowed for FTL from a planet of higher-tech but non-spacefaring aliens in exchange for a performance of Romeo and Juliet and teaching them the play. In the original canon none of the aliens are notable beyond the scope of one story, and that seems like a good way to do it: if you want to write in the world and want aliens for your story, go for it, as long as it makes sense that other stories aren't mentioning them.
Shellperson culture: Shellpeople as a rule know nothing whatsoever about their pre-shell life; this is presented as a security issue, and that's not exactly false - shellpeople are often in positions of high trust; making it harder to threaten or blackmail them is not entirely unreasonable - but it's mostly because Sunex wants their undivided loyalty for as long as they can have it. (Getting ahold of one's own records is scandalous; looking at another shellperson's records is incredibly taboo, a major breach of privacy.) In practice, shellpeople over the age of 30 or so are generally more loyal to each other; 'part of the Sunex family' pretty easily turns into 'part of the shellperson family' when they catch on to Sunex not actually being that great. This doesn't extend to financial help - a paid-off shellperson might offer a close personal friend a loan to get them out of a tight spot, but otherwise it's just not done and definitely not asked for - but in any other context they actively look out for each other, to the point of risking their lives for each other when necessary. This extends to brawns, too; one of the books has a bit where an ancient and falling-apart brainship with no communication capacity approaches and in fact nearly runs into a station, and upon realizing that the ship is a brainship, the station's brawn risks her life - in the sense of 'well, I'm going to run out of air two minutes before I'm done, but' - in an attempt to save the ship's shellperson, who may not even still be alive, and both the brawn and the station treat it as something that of course she would do, that's just how this works.
Another feature of shellperson culture is a relatively strong focus on music and especially singing. This is because of Helva, the most famous brainship, who was instrumental in the above-mentioned FTL energy source trade, and who was famous for at the time being the only ship who sang; now one of the only shellperson classes that doesn't directly relate to their eventual careers is a choir class, and every shellperson at least knows how good they are at singing and whether they enjoy it. (They're not particularly more likely to have either of those traits than a random person would, though.)
Cultural reaction to shellpeople: Most people are just barely aware that shellpeople exist, and have definitely never met one. As such, misinformation abounds, particularly about what shellpeople are (brain-in-a-jar; really fancy AI) and what they're capable of (people are all over the map on this one, but underestimating their personhood and awareness is a common theme). Xenophobia - often in the form of ableism - is common. Individuals who frequently travel through space are generally much, much better about this than people who don't, as nearly all space stations are shellpeople and they tend to make that fact quite a bit more obvious than city or building shellpeople. Among people who are knowledgeable about shellpeople, the opportunity to travel via brainship, or shellperson attention in general, is acknowledged as a luxury, and there are more than a few brainships who make a good living ferrying VIP passengers from place to place.
Names & designations: This world is far enough in the future for some linguistic drift to have happened, including to names; most names are noticeably related to 21st-century ones, but with some changes. (Surnames are largely unchanged.) Shellpeople don't know or use their surnames, and are variously recalcitrant about sharing their given names; for professional purposes they use their designation instead. Most designations are in the format [Brawn's given-name initial, or X][Brain's given-name initial]-[number], where the number is given sequentially by job type. So, Denika's designation before Ifan becomes her brawn is XD-1213, indicating that she's brawnless and is the 1,213rd brainship to ever exist; after Ifan becomes her brawn, it's ID-1213. Space station designations are instead in the format SS[Brain's initial]-[number]-[Brawn's initial if any].
Brainship features & description: A standard brainship has a central cabin that contains the brawn's workstation, a small galley (kitchen), various seating and viewscreens, and the titanium column that contains the shellperson's shell. It has eleven private cabins: the brawn's cabin and VIP cabin, plus nine standard ones that are half that size. Ships optimized for VIP transport or long-term habitation will instead have five large passenger cabins and one standard one (often repurposed), while some ships opt to split the standard VIP cabin into two. Every ship also has a small (standard cabin sized) but well-stocked medbay with servos that allow the ship to provide medical care to its inhabitants. In addition, brainships have a good-sized cargo bay with servos to assist in loading and unloading it and also a grav sled (open-topped roughly carlike flying vehicle). There's also an airlock and space suits available. By default everything is done in Sunex-standard Calming Grey, but it's not especially uncommon for a ship to spring for personalizations, especially in the main cabin, or just add decorations on top of it.
Techwise, a brainship is outfitted with cameras such that they can see every part of themselves, inside and out, and hear and speak in any room or outside. (They can also not do that, particularly in the private cabins.) They can control every system on the ship - for example, in one of the books a ship changes her brawn's shower temperature from cold to hot to encourage him to go to bed when he's been up for several days straight working on a problem. For security purposes, they have the ability to flood any room with sleep gas.
Further divergences from the original canon:
- Brawns having any input whatsoever into opening their partner's shell is not a thing; gross 1950s marriage-based concepts of consent have no place here.
- Similarly, brawns in general having a tendency to get obsessed with what their partners 'really' look like/wanting to touch them/etc is just not a thing; this makes no sense for a post-internet culture.
- to the stars spoiler: In the original canon the reaction to Hadrien killing his ship and taking one of her ovaries to have a kid with would have been roughly 'yeah, geez, we should have seen that coming, how did we miss that we should have been keeping an eye on that guy'; here it's more like 'Jesus Christ that's like not just killing someone but peeling their skin off and wearing it, completely around the bend, what the fuck, no way we could have predicted that it just doesn't happen'.
- Humanity did send out some generation ships before FTL was a thing. A few of them were brainships; none of them had any way of getting back. In the original canon it's ambiguous whether this was voluntary (we only hear about it from a modern station, who's vaguely appalled at the whole thing), and that's still the case here - it was voluntary in the sense that the shellpeople in question agreed to it, but questionable in the sense that it was leaning on their indebtedness - 'we'll pay off the rest of your Sunex debt if you'll come be the ship for us' kind of thing, of course that's going to get people who're more interested in no-Sunex than in yes-never-seeing-civilization-again.
- Original canon has the concept of 'high families' (sort of inherited informal aristocracy, there's more to it than just them being super rich), which I'm not particularly sold on but might make an interesting plot point sometime. *shrug*