Short Answer: Mongoose Core Rules
As far as ease of understanding the rules, the best written core rules are hands-down Mongoose Traveller (MgT). The supplements, however, are less generally well received. Central Supply Catalogue and High Guard are well received; the rest have lots of people who don't like each of them for various reasons. It does, as a line, suffer from early and significant errata. Traders and Gunboats is fine on paper; in PDF, however, the raster-based images make most of the deckplans illegible, and you can't blow them up, either. Some of the 3rd party supplements are excellent.
T20: Traveller's Handbook is very accessible to D20 players, but has major mechanical twists from standard d20, and is based off of the D20 3.0 SRD. Setting is 100 years prior to CT, but still, essentially CT in flavor; ship designs compatible with CT Book 5 designs. It lacks a few elements from the d20 core, so familiarity with d20 is a must.
GURPS Traveller (GT) is probably the easiest edition for anyone familiar with GURPS. Setting is 20 years after CT, but essentially still CT. Needs a GURPS Core Rulebook, and was written for 3rd Ed. GURPS Traveller: Interstellar Wars (GTIW) is set several millennia prior to CT and GT, and is for GURPS 4th ed. Note that the GT license is ending, and new products will not be available.
For Hero System Fans, the Traveller For Hero books are awesome - but they are
permanently out of print, and for 5th ed Hero. Setting information for CT, Mega, TNE, and T4 era play is provided in schematic. Needs Hero System Core Rules (4th, 5th, or 5th revised). Edit: Marc Miller has announced that there will be a Traveller for Hero CD.
Traveller The New Era (TNE) was fairly well written, second only to Mongoose in clarity, but is not mechanically compatible with other editions. Further, the changes in the mechanics also are mirrored with setting destruction. I don't recommend it on the basis of it's setting and rules being so skewed from the rest. For those not looking for the standard setting, it's a very playable game.
MegaTraveller has massive errata issues, and some clarity issues, especially with combat. While I love it, it should be avoided if looking for an easily accessible ruleset.
Classic Traveller has numerous clarity issues. It's very much "old school" - you will have a lot of interpretation to do when running it, and there are multiple options for every part of the game. The adventures, however, are easily adapted to any other edition, and so the CD is a recommend, even tho' the game isn't.
Marc Miller's Traveller (T4) is noted for unpleasantly large amounts of errata, and I found it extremely poorly edited. The rules are not hard to understand, but have some issues that make it impossible for me to recommend, including the tech book being badly mangled by poor edit/layout, and the core rules ship design being insufficient for the setting as presented. Setting is at the dawn of the 3rd Imperium, 1100 years before Classic is set.
T5 is a monstrously large PDF or Hardback. It is playable, as is evidenced by the number of groups successfully running it PBP/PBF, but has errata issues, poor organization, and requires extensive GM prep before play. It is easily the least accessible edition for new players, as it is tersely written, table heavy, requires significant changes in thinking for players of older editions, and is under revision to a new sub-edition by the designer. Edit: as of the T5.09 PDF, the readability is much improved. Still a lot of errata issues, but now it's mostly typos.
Traveller:2300, later renamed 2300AD, is not actually Traveller, in the sense that it shares neither rules nor setting with any other edition. It's an excellent game on its own, but is not as well worded as TNE nor Mongoose. The mechanics show many similarities to MT and TNE.
There is a T20 adaptation 2320; 2320 was excellent, allowing play under a Traveller ruleset. The adaptation book for Mongoose is by the same author as 2320, but is not as polished, nor does it advance the setting timeline like 2320 did; it does, however, have multiple support books.
Available editions in PDF from Far Future Enterprises direct on CD or via DTRPG: Classic, MegaTraveller, Traveller: The New Era, Marc Miller's Traveller (T4), T20, and the CD for T5.
Available in POD: CT's The Traveller Book is available from DTRPG in "Print on Demand" - the initial reviews of the quality have been good.
Available in PDF from SJGames: GURPS Traveller and GT Interstellar Wars (GTIW)
Available in PDF from Mongoose or DTRPG: Mongoose Traveller (all but most recent releases)
Currently in Print: Mongoose Traveller, T5.
Unavailable for various reasons: 2320 is unavailable at present, as are the DGP materials for CT and MT. Many smaller 3rd party companies' materials and the Dragon Magazine articles are unavailable as well.
Bias Warning: I was a playtester on T20, MgT, and 2320. Further, I administer the official Traveller forums.
Edit notes: As of 1 Aug 2015, SJG has announced the end of their license, and Marc has announced that he's putting Traveller for Hero on CD.