I'm hooking lights in my shed and I want to know if I can go from a 10-3 wire down to a 12-2 wire. Is this allowed or will it cause problems?

The size of the cable usually indicates the size of the overcurrent protection or circuit breaker. #10 awg cable is typically protected by a 30 ampere circuit breaker. Connecting #12 awg cable to a 30 ampere circuit would be a code violation as #12 should be protected by no more than a 20 ampere circuit breaker.
If you change the circuit breaker to a 20 ampere, then you can use the #12 awg...