"Rhythm question" by a beginner

[indigotree]

Hi!

I have just decided to start playing the piano, and am wondering alittle about rhythm.

My assumption is that the speed a piece is played in is decided by the pianist and not the sheets. So if I decided to play quarter notes one click per second, the eight notes would be played at half a second. And if I chose to play the quarter notes at half a second, I would have to play the eights at a quarter second, and then I surely would have some problems with the sixteens... But is it correct that this speed is relative and not set up by the sheet?

Thank you very much for taking the time.