The essentials are all there in the core rulebook. You don't need anything else, because all the rest of the series are just gravy.
However ...
While the role of Mercenary is optional, as is Scout, both being as they are sourcebooks dealing with a particular career type, one essential book is already out. High Guard is the essential guide to starship design, a valuable addition to the core rulebook's starship design rules. High Guard enables you to create vessels ranging from the likes of X-wing fighters through to mighty battleships. Indeed, it can be said that the basic starship creation rules alone might not be sufficient to design some of the more exotic types of starship you are likely to encounter, such as million ton sublight generation ships and solar sail pleasure yachts.
Will your campaign have a heavy Belter emphasis and presence? The adventure Beltstrike has the Belter career paths, and a close look at that particular career and at Belter culture. Asteroid mining is given a good, long look, and asteroid habitats are described in Beltstrike in detail (something to consider if you have a capital world set in an asteroid belt like Glisten in the Spinward Marches).
If you are planning on having your characters thrust into the back end of the legal system as convicts in a prison setting, Prison Planet is your best bet here. This adventure details a fully working prison with optional Ancient alien base, for a plot twist or two. Prison Planet explores prisoners' culture, gang warfare, explains the Reputation character development system, looks at how to run gangs as collectives using a modified version of the Reputation rating covering each gang as a single entity, and it goes through a list of useful tech and equipment, including narcotics (if you're running a law enforcement story, you might want to look at that list as inspiration).
If your characters are going to be thrown into an "Across The Bright Face" - style survival story, Scout is your best guide: this book takes a long look at what Survival really means, introduces the Stress rating (a system revisited in Prison Planet) and outlines exactly how, and where, Survival skill is most useful to wilderness-exploring Travellers.
Every book has at least one useful article in it, whether it's something about the real world elements which a character might want to go prospecting after (either asteroid mining or mining on a planet like Tarkwall in Prison Planet) or lists of equipment and vehicles which scout service personnel can call upon, such as ground effect flycycles and ManPac sensor packages (think modular tricorders, and you'll have a fairly good idea).
Psion? Hasn't come out yet. Don't worry. If you like your scenarios to ramp up the weirdness an order of magnitude, this looks like your best starting point. Apparently mental trauma is described in this book, so if you want your characters to enjoy your very own home brewed flavours of madness, you might want to wait till Psion comes out first.
Oh, and Agent (a book that's a long, long away compared to Psion) is being touted as the book for all those high tech hijinks and Q Division shenanigans that serve to blur the line between spy thriller, SF and cyberpunk. Mission: Impossible in space with a hint of Burning Chrome? That, I believe, is Book 5, publication date TBA (but eagerly awaited).
Vehicle guides are in the pipeline, as is the World Builder's book (which allows you to detail every hunk of rock and gas giant in your systems) and, I believe, another adaptation of a CT adventure, Murder on Arcturus Station, which will allow you to plan and run a classic Whodunnit murder mystery scenario, where logic and investigation will be the characters' strongest weapons.
They're all absolutely essential, and yet they're all also entirely optional. Each one gives you one small facet of the world of Traveller according to Mongoose, but you only really need the core rulebook to run a full and complete Traveller game.
The core rulebook, and High Guard. (You'd need High Guard to fathom out the ship designs in Traders & Gunboats and also Fighting Ships. If you're looking at these two books, High Guard is essential).
Ite. Missa est.
