Dragon Age (at least Origins) + limited time = not a good idea. It's a pretty extensive game, particularly if you like to explore everything. The game pushes you to explore everything, since the core quests are pretty long, and there are a good amount of sidequests, particularly if you want the right amount of levels and whatnot.

Limited money, and still interested in playing? The Ultimate Edition of DA: Origins gives a lot of value for your buck, because it has essentially all downloadable content in two discs. It contains Awakening (the game's expansion) for extra hours of play, plus a few scenarios (one where you control the main enemies of the game, one where you learn about the story of an NPC, a "stand-alone" scenario and what's essentially an epilogue) for extra playing time.

Can't say much about DA II, except it doesn't has a Ultimate Edition (as far as I know of) and most people believe it's an inferior sequel. It has a fixed main character (you can customize him/her, but it's mostly a human), and the story is a bit more contained than the first one.

The third game in the series is coming about one year from now, so no one can tell you if that's the game you should play. In any case, if you're decided to play a game, it's best to start with the beginning for a wide variety of reasons.