Really, only the main rulebook is essential. It has rules for aliens, animals, creating subsectors and worlds, some spaceship designs (and a system for making new ships).
The other books let you dig deeper into different campaign styles and character generation for those topics. High Guard (ships) and Mercenary are more "crunchy", while Scoundrel (folks who aim to misbehave) and Dilettante (everything from Star Wars style "important people trying to save the realm" to "travelling rock stars and their adventures") are more free-wheeling.
The alien modules are for either running a non-human campaign or giving the Referee a deeper understanding of an alien culture they may want to draw antagonists (or even allies) from.
The supplements generally give you catalogs of ships, vehicles, weapons, critters, etc. It's mostly stuff you could generate on your own, and in some cases some new or expanded rules.
The setting books (sector books) give you a "map" to play on if you don't want to generate your own. Generating a subsector is fun, but detailing the subsector takes a while. Though in fairness, the sector books generally only really detail a few worlds per subsector. The alien modules also include sectors or subsectors (depending on the size of the various aliens' domains) which are useful in the same way.
[It's worth mentioning that between travellermap.com and the traveller wikia site, you can get quite a lot of standard Traveller information. I'd look into Reaver's Deep in addition to the classic Spinward Marches. There's even a web-zine
https://sites.google.com/site/reaversdeep/ which is writing up Reaver's Deep in detail.]