Vanquish is a fast, hyperkinetic 3rd person shooter, combining the best stuff of run and gun with cover type shooting games. That said, the story is more plot hole than plot, and pretty poorly written. There's no POC and it's all about grumbly white men in power armor, and the hot blonde woman in a mini skirt, and evil President Hillary Clinton. More on that at the end...
So, gameplay is really cool - the levels are linear, but the pop and drop rooms and layout of cover is really well set up. Your power armor lets you power-slide, which is crucial for moving from cover to cover, which you'll need to do as the enemies become smarter about flanking and rushing you.
The limit is, power-sliding too much overheats your suit which leaves you slower and weaker while it cools down.
The game does an excellent job of ramping up difficulty without just adding more enemies. You'll find the same enemy type you fought a few levels ago is a lot tougher in tighter spaces, or different environments. There's a few gimmick levels, but they don't feel too gimmicky- fights on the back of moving trains, flying vehicles, zero-G levels, etc.
Story-wise.... eh. In the future, Russians take over a US space colony and use it's microwave beam to fry San Francisco. You play aforementioned grumbly white man Sam, who is using prototype power armor to help free the station and rescue the scientist who helped build his armor and the station.
Basically, the rest of the story is terrible action movie cliches which don't make sense in their context. Lots of important things go left unexplained: Why do the Russians have such awesome robots? How is it they can teleport things and you can't? "Temporal Disturbance" is mentioned- the Russians can travel through time? I don't know!
At the end of it all, the non-spoiler spoiler is that Evil President Hillary Clinton was working with the Russians on this, and shoots herself in the head, and you get to her laid out with the gory head wound.
...I'm not sure what hate-on they have for Hillary to take it that far, but clearly there's deep issues at hand with the writer.
Overall, the game has really fun game play, and shitty story. Which, I guess is classic Sega. I recommend it if you're into hyper shooting games, as the gameplay is excellent. I don't recommend it for anything else.