Assume that I played an ability of Garruk Relentless and I successfully transformed him, do I get to play his transformed ability or do I have to wait for the next turn?
Gatherer says that
You can’t activate a loyalty ability of Garruk Relentless and later that turn after he transforms activate a loyalty ability of Garruk, the Veil-Cursed.
So you can't. The actual rule in play here is 606.3:
A player may activate a loyalty ability of a permanent they control any time they have priority and the stack is empty during a main phase of their turn, but only if no player has previously activated a loyalty ability of that permanent that turn.
(Emphasis mine.) Transforming a permanent does not make it into a new object, by rule 711.12:
When a double-faced permanent transforms, it doesn’t become a new object. Any effects that applied to that permanent will continue to apply to it after it transforms.
Therefore, if you have already played one of the loyalty abilities of Garruk Relentless this turn, then after he transforms into Garruk, the Veil-Cursed he is still the same permanent so the rules forbid you from activating any of his loyalty abilities.