It really depends on your interests as a gamer. I would personally recommend playing all four canon games (Tactics produced a hearty 'meh' from me, and BoS... just no), but I realize that not all of them are for everyone.
1 and 2 are brilliantly written, with an expansive world, harsh tactical combat, etc. but a lot of modern gamers hate them for the crappy graphics, turn-based system, and harsh tactical combat.
Fo3 probably has the lowest writing quality of the bunch, but it's still a very good game. The GOTY edition also means that you'll probably get the most fun out of it for the least money. The DLC was pretty hit-and-miss (I loved Broken Steel and Point Lookout, Operation Anchorage and Mothership Zeta were decent but not as good as the main game and seriously unbalanced things when you took the gear they gave you into the rest of the world, and the Pitt was just crap), the main quest was pretty railroad-y and too short, the voice acting was crap and the animation could have been better, and it's still pretty buggy, but overall it's a very solid game, and probably the best starting point for a modern gamer since it's kind of disconnected from the main continuity of the others and it's closer to most modern games.
New Vegas is my personal favorite; it takes an ungraded version of the Fo3 engine and returns to the quality of writing, depth, and storyline of the first two. Despite what you've probably heard it's not horribly bugged (anymore); it's definitely more stable than Fo3 at this point. The DLCs (at least the first three, and Lonesome Road isn't that bad) are wonderful, but a bit expensive to get all individually at the moment. Mechanically it's got a better level of realism and immersion than I've seen in a good long time and it's very well balanced, world-wise it's expansive and filled with realistic characters and interesting factions that interact realistically, and story-wise it's complex, well-written, and has a lot of flexibility.
One important thing to note though: unless you're completely skipping Fo3, you should play that before NV. Fo3 is a good game, but going back to it after New Vegas all I could see were the things the next game did better; I couldn't just enjoy it for what it was. Also, going through Fo3 will probably give you time for New Vegas GOTY to come out.