Arsenal are definitely the team to watch, though I'm a Gooner so am perhaps a little biased. And for the same reason, you should absolutely avoid
alanR's ridiculous suggestion to go watch Tottenham.
How long will you be studying in London, and where? It always helps me to feel part of where I live by getting close to the sporting teams so you might want to think about picking a team to more actively support.
As
alanr says, Fulham are generally quite easy to get tickets for, especially for games that don't involve Man U, Arsenal, Chelsea etc.
I'm assuming you are American, yes? If so, Fulham is probably a good choice. Nice little stadium in a pleasant part of London. Good crowd of generally decent people. Whilst the game has been largely cleansed of extreme behaviour, what passes for "normal" in a British football crowd could shock most Americans who are perhaps more used to the somewhat sanitized atmosphere that is prevalent in most "franchised" professional sports. Loud profanity is pretty common, irrespective of whether kids are nearby and yes, this even includes that little word that seems to still genuinely shock people in the US (I always tell them it's a term of endearment where I come from, which is at least partly true.....). Anyway, Fulham fans are less likely to exhibit neanderthal behaviour than some of the other clubs (and yes Chelsea, I'm looking at you.....)
If you're after a more genuine experience, try West Ham. Even though they are likely to be playing Championship football next season they will still get big crowds but getting tickets for the Hammers is normally straight-forward enough. A very working class club in a working class area of town. The fans are a little more rough round the edges, but I've always found them OK.