Good band suggestions?Watch
new bands (90's onwards): Riverside, Opeth, Tool, Airbag
oldies: King Crimson, Camel, Genesis (also Peter Gabriel's or Phil Colin's solo work), Marilion
If you like Radiohead then definately check out Massive Attack (start with Mezzanine album) and Archive. Also check Air.
If you like Nirvana then you MUST know Pearl Jam (go for "Ten" album), possibly also Soundgarden and Foo Fighters (I don't really like them, but they are related to Nirvana).
Finally, if you are interested I can give you links to a bunch of Polish rock bands (or rather their albums on YT), since I'm Polish and always regret that they didn't become internationally popular due to the language barrier... But that's only if you are in interested in going a bit beyond the Anglo-Saxon world
EDIT: I came to my mind that you can also try Atoms for Peace, kinda avant garde, like a musical experiment, but it was started by Flea (RHCP) and Thom Yorke (Radiohead), so theoretically its an ideal choice, haha.
An international success at the 60s and 70s, this pop-rock group even made it into the Vocal Group Hall of Fame before the Beatles.
Couple of songs from them to listen to:
Big Girls Don't Cry
Walk Like A Man
Can't Take My Eyes Off You
fall out boy
panic! at the disco
lower than atlantis