I actually am not too interested in the Third Imperium setting either. As someone who just a week ago never knew that Traveller existed, the amount of information is just too much for me. The combination of this huge established setting to sift through and my own inclinations for creations (I'll admit it I am a homebrewer at heart when it comes to RPGs and settings) I don't have much need for the Third Imperium info.
In the other thread this one forked from, you mentioned you wouldn't touch the Mongoose version of Traveller. Any particular reasons why or just because it comes without as many 'strings attached'? Is the resolution system in CT still 2d6+abilites+skills+/-modifiers?
Actually the "system" in CT is more open than that. For combat, the base roll is 8+, and you get bonuses and penalties based on whether your Dex is high or low for that weapon (example: the shotgun has a low minimum but requires a high value to get a bonus), skill rating, range and the armor the target is wearing.
For things outside of personal and ship combat it is left rather freeform. The expectation seems to be that the Ref will come up with target numbers on the spot; your roll will usually be modified by your skill level and maybe by your attribute if the Ref says so. Some Refs use a universal system (like the "6-8-10" diff ratings, with standard attribute bonues) whereas others may just do spot rulings. Either basically works fine.
On Mongoose, I have heard rumors that there is some rules weirdness, like being able to armor up civilian ships to be impervious to missiles or something like that. Basically every Mongoose product I've ever bought has had crazy, obviously untested stuff in it that blew my mind with how stupid I thought it was. So I've sworn off buying Mongoose stuff. I actually have no idea how married MongTrav is to the Third Imperium setting.
Anyway, I like the bare bones basic rules and minimalist feel of the 3 LBBs. Then it can be as fancy or as sparing as I choose for it to be. If you do want to go that route, you can get the 3 LBBs in a few different ways. QLI (the publishers of T20) did a reprint that was about 10-12 bucks, but I've heard that their website may be dodgy as far as ordering goes, so you might go with a retailer (online or otherwise). I think this book may be available through Mark Miller's company FFE (Far Future Enterprises), and the product code for it is FE 000. Then there's FE 001 "The Classic Books", which gives you Books 0-8 (the 3 LBBs are 1-3; 0 is an introduction). The later books have interesting stuff in them, but I use almost none of it personally.