I'm looking for a game that features fast character movement and explosions, lots of explosions, like explosions as a main attack.

Any suggestions?

Shovel Knight comes with the expansion Plague of Shadows which satisfies all of your criteria.

Unreal and Quake games?

Doubtful because the explosions are ammunition limited. I want a game that gets the hell out of my way. Also, the radius of the explosions is tiny in those games. I didn't want to be too specific with my request because I'm not optimistic about getting what I want. But, I'm curious what people will suggest. In the end I'll probably have to resort to Skyrim or Fallout 4 mods.

or [url=http://store.steampowered.com/app/105420/]Ms. Splosion Man.

Then how about good old Bomberman? :P

Silent Bomber! The game that's not really silent because the protagonist is always whining about things.

Especially Act Zero.

Batman Returns (Sega CD)?

There's a lot of explosions and fast movement (for most classes) in Team Fortress 2. If you play as the soldier then your main attack is a bazooka, which you can also use to rocket jump. If you play as the scout then you can cover the length of the map in mere seconds, all the while pulling off some impressive maneuvers.

Asteroids

derrr, yep, thanks tinyE for the correction, of course Asteroids

giant space objects zooming at you and BOOM BOOM BOOM explosions destroying them, fast paced destruction via powerful missles at objects miles in diamter! exactly what the OP is looking for!

Don't you mean Asteroids?

Or 'Splosion Man (of which Ms. Splosion Man is a sequel), if XBox 360 is acceptable. Like the Saints Row series, the first game is only available on console.

Saints Row IV

Once you get enough superpowers you can cause a ridiculous amount of explosions with your powers, plus you can run super fast (literally). You have an explosion element that you can attach to your superpowers that causes your explosions to make the victims of said explosion explode. :)