Choosing two genres to describe the game I would probably have gone with Platformer and Shooter instead.
There are a few Adventure elements in that you explore many locations thoroughly, and there are Adventure-style logic puzzles (although I don't see these as a particular strength of traditional Adventure games). But in my opinion there's not enough narrative to call them Adventure games.
Apart from the jumping and climbing, the dangling and shimmying, the gliding and dropping that definitely make it more 3D Platformer than anything else, there is a lot of gunplay though. Due to auto-aim, the difficulty is not really in aiming, but in evasion and how to deal with multiple aggressors, stuff like that. There's enough shooting though that I would call it several partds Third-Person Shooter as well.
Maybe 60% platformer, 30% TPS, 10% Adventure =) It is definitely fast-paced and very arcade-y though, so if you're looking for a narrative-driven game with the occasional action scene, Tomb Raider isn't it.