He jumped genres many times in his career. The only consistent theme is that it's weird.
To summarize way too much: he started out more rock-focused, then moved into more of a jazz-rock fusion, then started bringing in more classical elements near the end of his life. Throughout all of it there's a strong experimental/avant-garde bent in both the music and the lyrics, and it can get extremely complex at times.
If you're into Queen and considering Rush, you should check out the "Shut up and play yer guitar" compilation albums of his. They aren't albums so much as exhibits to showcase the kind of insane guitar playing he was known for.