Action-Adventure, Thriller, Satire, Religious Picture
Action-adventure is the name of the game and Raiders of the Lost Ark did it so well it practically defines the genre these days. They even added it to their ad campaign for the second film: "If adventure has a name, it must be Indiana Jones." They're claiming their turf and they ain't bein' shy about it either. If you had to pick one genre for the movie, it shouldn't be a difficult choice.
Underneath that umbrella though, we can find signs of a few other genres poking up here and there. The thriller, for starters: a genre related to adventure films but only indirectly. Fear-inducing threats like snakes and spiders become part of a near-constant stream of peril, as do the mummies, cannibals, and a really serious smackdown from the Creator of All Things. If your pulse doesn't race at least a little bit in this film, you might be dead, which means it has at least a little thriller in its DNA.
Satire plays a small but important role as well. While Raiders really wants to honor the serials that inspired it, it also wants to poke fun at some of their sillier clichés. Hence we get Indy beating up a guard and taking his uniform, only to find that the uniform doesn't fit. Add to that the fact that he gets beaten up almost as much as he beats other people up (something that never would have happened in the serials) and you have a mischievous side to the movie that isn't afraid to poke fun at itself a little bit.
Finally, no story featuring the Ark of the Covenant can escape at least some of the trappings of a religious picture. Raiders lends its Ark a sense of awe by lighting it in a golden halo at all times and putting John Williams' eerily reverential theme on the soundtrack. The end features God directly taking part in the proceedings, which by definition qualifies this for religious picture status.
That's a bit of a cocktail from a genre perspective. The good news is that it blends all those ingredients perfectly, creating a final film composed of pure awesome.