Thank you all for your opinions and suggestions. My only concern is from a consumer standpoint, buying from the core rulebooks usable for any hard sci-fi or sci-fi opera, into the setting of Traveller itself is a very rough and unclear way. I think I will start with the Aliens of Charted Space and as others suggested Behind the Claw will be the first setting book I will buy into. But beyond this, it is very hard to tell do I need The Solomani Front, do I need Third Imperium, do I need Glorious Empire, etc. I don't even know if the Vargr have their book yet, and I have just recently discovered Great Rift exists and for the vast emptiness of it, there is a whole boxed set. It is really hard to tell what to buy and where to base my sandbox for my players.
It's hard to know without a little knowledge of the setting, but I think the marketing for each book gives you a good sense of what's in them.
Behind the Claw is the traditional setting: the Spinward Marches, and it has Deneb sector as well. Frontier area with lots of different factions and aliens. Plus, if you dig up old material from previous incarnations of the game, you'll find cool bits of lore and complete adventures that you could convert and use there.
The Third Imperium features the Core, where the Emperor sits upon his throne, the place where the Imperium started over a thousand years ago. It's a more settled, generally higher tech, political region, with lots of dystopias and several key worlds of Traveller lore. There will be a campaign added for this region next year.
The Deep and the Dark contains two sectors where the presence of the alien Aslan is an important part of the narrative. It contains their homeworld and the polities that border their region. Lots of cool minor alien species are present in both sectors if you like the idea of a diverse and complex region to run your game.
The Solomani Front is a politically tense region that the Imperium largely occupies, but in which the Solomani, descended from Terra, hope to regain their homeworld and dominion over the region. Like the Core, it's got consistently higher tech, higher populations, and features a politically rich background where border conflicts and tension would be rife. The presence of Terra is a big part of the appeal, but the Solomani are a disjointed, often racist lot.
There's nothing stopping you from running games in more than one sector, but I realize the cost of all those books could be prohibitive. It is fun to jump around and explore different regions, though, if you're into published settings.
As others have suggested, you could build your own, which probably fully half of all Traveller players do.