If you're looking toward a singular goal with a couple of different endings, Fallout 3 is the way to go. (It's also a good way to break you into the first/third person, open world, action RPG.) Not quite so much to work with. Simpler than Fallout New Vegas.
(If you prefer a bit more sword and sorcery....try the Elder Scrolls games, starting with Elder Scrolls IV: Oblivion, which was my first Xbox 360 title...and still one of my favorites to this day.)
If you want a somewhat more involved experience, a smoother first/third person shooting experience, and more options of how to end the game, enjoy crafting: Fallout New Vegas is the way to go. Lots of weapons (from melee to firearms) can be customized. Also, you can try your hand at Caravan, which is a card game played by several wasteland denizens, and of course, get to Freeside or New Vegas, both of which have gambling establishments. You can play poker, slots, or my personal favorite, Blackjack.
Also, certain factional outfits that you wear can have interesting effects on other wasteland inhabitants....i.e., wear the wrong faction colors, and you might find yourself as a human target. But, sometimes....sometimes, such outfits can also be disguises to infiltrate factions. Remember:
sometimes.
The stories are standalone....different characters, just in the same universe. Both games provide some history about the Fallout universe. The DLC for both are really good, but my favorite DLC is still Mothership Zeta for Fallout 3.
Fallout 3, you play the son/daughter (depending on your gender choice) of a scientist who went missing shortly after your 16th birthday.
Fallout New Vegas, you play a courier (again, choice of gender) who was shot and left for dead because of a small device that could change the balance of power in Nevada...and now, caught up in the middle of a big war between major factions, you're lookin' for some payback.
