Start with the most simple. pedal and shift by pressing directly on the cable arm of the FD. Does it move smoothly? Does it move far enough? If not, what's holding it back. It's easy enough to figure out which limit does what by noting how far the FD moves in each direction, then tightening one two turns and seeing which direction is now restricted. (the other limits the other direction).

Note the limits don't actually move anything, they just determine the end of the range of motion in that direction. While the cable is off, adjust the inner limit so the chain stays on the inner (innermost of 3) chainring when you let go, and shifts to and stays on the outer chainring when you press and hold the arm. You might need to fine tune later, but you now have the FDs range within reason. Now attach a new cable, and adjust it's length so the lever positions correspond properly to the various FD positions.

Usually you need to pull out all the slack while on the innermost chainring, but this can be difficult so use a pencil or something similar to jam the FD about 1/8" out from the inner ring. Pull up the cable, tighten the pinch bolt, and remove the pencil and the FD should drop back with the cable fairly tight.

BTW- if your bike does not have a barrel adjuster on the lever or frame, buy an in-line adjuster and install it when fitting the new cable. That will make fine tuning the FD cable length much easier.