My kitchen sink P-trap nut is stuck with old caulk or putty. What's the best way to loosen it?

That's usually hardened plumber's putty or caulk binding the nut to the pipe. Try scoring the sealant around the nut with a utility knife first, then use a pair of channel locks. If it's a plastic trap, be careful not to crush the nut; older plastic gets brittle and might just need full replacement if it won't budge.