There's probably people with a more encyclopedic command, but I'll start out by trying to give some context to the setting info in the rules sets you may see out there:
Digital copies of pretty much everything I talk about are available from
Far Future Enterprises.
There's also a Traveller Wiki with a lot of good information. Here's the entry on the
Third Imperium.
Classic Traveller (CT): The original rules set from the late 70s was setting independent, and what would become the official setting (sometimes you'll see it referred to as the OTU, Official Traveller Universe) of the Third Imperium came in dribs and drabs. The wargames Fifth Frontier War and Invasion Earth give good detail on two large conflicts in the classic period. The adventures are another good source.
CT is set in the Classic Era, which is roughly the years 1105-1116 of the Third Imperium, right up until the assassination of Emperor Strephon.
MegaTraveller (MT): This edition dates to the mid-80s, and it introduces sweeping mechanical changes, and also it shakes up the setting by having Archduke Dulinor murder Emperor Strephon. This leads to the fracturing of the Imperium into several factions and a lengthy civil war. It covers the years 1116-11130, and things become increasingly bleak.
The rules are much more closely tied to the setting than in CT, so the information about the setting is easier to find. The Imperial Encyclopedia and the Rebellion Sourcebook are good general background information. Hard Times and Arrival Vengeance do a great job of showing the deterioration of the Imperium.
Traveller The New Era (TNE): This edition is the most controversial, because it uses a radically different rules set (not very good one, imo) and huge changes to the setting. TNE takes place in the year 1201, 70 years after the civil wars are ended by the inadvertent release of Virus, a weaponized, self-aware AI computer virus. It was a super weapon developed by Lucan the Black, one of the faction leaders from MT, but it gets out and essentially destroys most of known space. TNE depicts rebuilding efforts decades later, or an alternative campaign can be set in a small remnant of former Imperial territory that managed to stave off the worst of Virus.
Many, many, many Traveller fans do not like (or even hate) TNE. For me, I quite enjoyed the post-apocalyptic mash up with Traveller, and I thought the way Virus was presented was interesting. The book Survival Margin chronicles the terminal decline of the Imperium through to the final release of Virus. The Reformation Coalition Equipment Guide, though meant as a gear book, has a lot of great tidbits relating to the setting.
Marc Miller's Traveller, aka Traveller4 (T4): Though much maligned for errata, print quality and various errors, this edition was interesting as it depicted the dawn of the Imperium, over 1000 years before the classic Traveller setting. Milieu 0 and the Aliens supplements have good setting concepts.
Despite being called Marc Miller's Traveller, he wasn't directly involved in making T4, from what I understand.
GURPS Traveller (GT): Published during the early 2000s, GT takes place in an alternate timeline where Strephon is never killed. Some of its books are very poorly regarded, like Behind the Claw. Others, such as Sword Worlds, were favorably received. They published some really great deckplans. YMMV.
Traveller D20 (T20): Traveller adapted to the 3rd edition D&D D20 system. Personally, I don't think D20 is great for sci-fi, but if it works for you, great! It's set in the classic era, but it details a previously under-developed area of the Imperium, so there's a lot of good information. The core book and Gateway to Destiny are both solid for setting information.
Mongoose Traveller (MgT or MgT2 depending on 1st or 2nd): A very polished, professional and accessible version of Traveller. Mechanically it's based on CT, though it has more features you'd expect from a modern rules set. I only own a few books, so I don't know which are best to get the setting.
Traveller5 (T5): The most recent edition put out by the original author, Marc Miller. It's kind of his
magnum opus. Though questions proliferate throughout the Traveller sphere as to how functional it is as a game, it is definitely incredibly detailed from a mechanical standpoint. I don't actually own T5, so I can't say much about the setting information.