Yeah, but you can't go super saiyan right away..You have to raise you Power Level, which are the white numbers in the blue bar. Goku starts at 750,000, so you have to raise the number by fighting or dragonballs (which is rare to find a public serber with esf dragonballs)
Once goku's powerlevel is at 1,750,000, press Z and an animation will instigate and after that, you will be a supersaiyan until you run out of ki, die, or manually decend by pressing "x".
Only goku can transform at 1,750,00 powerlevel (pl), so other characters have different power level requirements which can be read at the character select screen.