Put a task to the background with bash

I know that you can start a background job with Bash doing foo &. However, the best way I know to put a foreground job to the background is to do Ctrl+z to pause it then bg 1 to resume it in the background.

Is there a faster way? Some Ctrl+Something key combination I'm not aware of?

No, there isn't. The terminal reinterprets CtrlZ as SIGTSTP independently of bash putting the process into the background. See the susp option of stty as well as the signal(7) man page.

You could bind a keystroke to bg (no argument means the current job) then you could do

Ctrl-z Ctrl-Something

So it would be just two keystrokes. You wouldn't want to use bg 1 in case there are more than one job present.