I'd say just the Core Book is essential.
I'll say it twice because I'm going to ramble about other products but they're very much secondary. It's really a toolkit for building your own game. They don't give you starships, they tell you how to build them. They don't give you a setting and worlds to explore, they give you systems to do it yourself. Tables for everything from generating creatures to patron (mission) encounters. Around it all is wrapped the campaign theme of merchant trading and mechanics for that are included. Firefly, really. But before there was a Firefly.
What jumps out at me as being missing are systems to create alien races. You've got a handful of stock races from the OTU but many simply aren't interesting, to me, or aren't what I'm looking for. Ever since the Cantina scene I think many roleplayers want serious variety when they think alien.
Flynn's Guide to Alien Creation from Samardan Press is a third party product but it seems to work pretty well. The GM can eyeball traits and qualities to create a particular alien race or use a random system that works with the concepts of the animal generation/traits system in the core book to 'evolve' a new race of your very own. Some details are left blank, detailed cultural traits for example, but you'll have a basic idea to work with that is a nice mechanical fit.
I've got the Robots and Cybernetics books and like Flynn's Guide they do seem to fill in some big "sci fi" gaps. I haven't spent enough time with them to vouch for quality and there seems some debate on that. Likewise, as mentioned, the new Vehicle book will probably be quite practical and handy.
The Central Supply Catalog feels too weapon fixated to me, I'd like a more interesting variety of goods and services held out to the players and not just upscale killing devices and body armor, but it does give players things to want. And that can motivate them.
After that, if you're wanting to double down on the Firefly/freetrader feel of the core book, I'd go with Merchant Prince (for reasons mentioned above and many more) and Scoundrel (because this is the fun, non-spreadsheet, side of things like theft and smuggling and piracy along with a colorful list of odd jobs folks might do) and 760 Patrons (tons of adventure seeds). Voila.
1001 Characters is also handy but creating NPCs isn't super hard. You can eyeball career tables and get an idea of what a typical grunt or goon might be able to do. We're not talking D&D stat blocks here. The advantage, though, is that they'll be ready to go with skills, gear and even a touch of biography if you want to improvise some RP around them.
Also I'd bet that the upcoming Campaign Guidebook will focus on the core freetrading and adventuring mechanic of the core book. In fact, that will be the book that defines which titles are and aren't essential. If it's referenced by the CG, you probably want it, if not it's optional. This is supposed to be a tool that all but automates adventuring (to the degree a GM wishes).
If you want to do something else, well, that's what books like High Guard, Agent, Mercenary and Scout are for. They offer alternative ways to structure a campaign instead of the freetrader/adventuring model. How well fleshed out a job they do of it varies from what I've read. I haven't actually used any of them yet. A truly nonessential, but fun, variation is Dilettante. Seriously. A whole campaign model around the rich and famous, too much money and lots of intrigue and decadence. I can easily imagine a campaign. One part HBO's Rome, one part Entourage, and lots of Caprica on the side.
As for settings? Spinward Marches is good, only thing better is making your own of course, and if you really want to stick with it there are many other products that slot right into it. I'd also get the standalone sector map. It really helps visualize how everything fits together. Sector Fleet is also very useful here too. After that, Spinward Encounters and Starport both have good stuff to offer. Two campaigns are set here as well, Tripwire and Secrets of the Ancients. I can't vouch for the latter as I don't have them yet though.