FFE for the win!

With FFE's reprints you get more bang for the buck.
Classic Traveller is time-tested by over 30 years of play, as opposed to MGP's "new kids on the block" which may not hold up as well under a similar amount of intense scrutiny.
Core: Mongoose
I would suggest using the core rules book from Mongoose, because it's "accessible": the players (if they want) can easily order it from game stores, bookstores, Amazon, etc. It has everything you need to play Traveller, and it's a physical, hardcover book you can touch.
If you love Mongoose, then you'll want to pick up more of it. Otherwise, you want to look for cheaper alternatives -- in which case, why buy one book when you can buy an entire library on PDF? In which case I suggest:
Resource Library: Pick One
Then, I would suggest buying a FFE CDROM for a vast and affordable wealth of companion material.
MT CDROM. For concise but well-organized and usable background material, buy the MegaTraveller CDROM, whose Imperial Encyclopedia and Referee's Companion together give you enough background material to keep you going forever. I believe the MT CDROM includes a starchart of the Spinward Marches as well.
CT CDROM. For in-depth background material plus a large number of adventure ideas split into several volumes, buy the Classic Traveller CDROM. I believe the CT CDROM includes a starchart of the Spinward Marches as well.
Extensions: Traveller 5
Later on, if you want to start experimenting and extending the game system, I recommend the Traveller 5 CD. It covers a wide range of usefully expanded rules, and like most Traveller systems is modular enough to pick and choose what you want to drop in.
I recommend waiting until the 1.0 release, unless you're a playtester at heart.