How would climate behave on a tidally locked planet? Would there be zones like the ones on earth or not?
It depends on how oceanic the planet is particularly on the dark side. If there is a large continent on the far side, ice will pile up on it, stealing water away from the dayside. There won't be much weather. Just desert on one side and a big mass of ice on the other. If it's mostly ocean then there will be a dark side icecap and probably a perpetual hurricane that stays centered near the west pole with strong winds and ocean currents exchanging heat between the dark and light sides.
The climactic zones would exist in concentric circles based on how far they are away from the east pole (the bit of the planet closest to the sun).