First, let's check everything BUT the derailleur. Check the cable. Make sure it's still tight when the chain is on the smallest chainring. Pull it to be sure it slides and rebounds freely. A bottom-pull derailleur with cable-housing (where the cable goes from the derailleur into a sleeve) is especially prone to sticky cables. See the section on cable cleanout and cable replacement.
Quickly check that the derailleur clamp is still in the original position, aligned properly so the outside edge of the cage is parallel to the large chainwheel. Check the height. The bottom edge of the outside edge of the cage should lie about 1 to 2 mm above the large chainwheel when you push the shift lever so bring the cage over top. The derailleur should sit just high enough to allow shifting into the big chainring. See the section of derailleur replacement for adjustment of the derailleur clamp.
Do you need a new derailleur? You can waste a lot of time trying to tune a derailleur with worn-out hinge mechanism. You should buy a new derailleur if: (1) The cage won't hold its shape. (2) The cage can be moved and forth more than a whisker when you wiggle it with your fingers. (3) The mechanism sticks or catches despite cleaning and lube.
Now let's start the tuneup. Loosen the cable.
If the chain rubs on the outer side of the plate while in the middle cogs (rear) and small ring, move the limit screw so the front derailleur cage is closer to the chain while in the small-ring, big-cog combination (further away from the bike frame).
Also, if the chain tends to fall off the small ring during downshifts, adjust the low-gear limit screw to move the end-position of the cage towards the bigger rings.Shift up and down several times. If the down-shifting (going from big to middle ring) tends to skip the middle ring, increase the high-gear limit (allow the cage to move further over onto the big chain-ring).
Toeing in the cage: Hot Tip. If you have trouble with sluggish shifting, you can toe-in the front of the derailleur cage. (Typical symptoms? The front derailleur barely transfers the chain to a larger ring, yet hesitates or fails to drop the chain back to a smaller ring on downshifts. You tune the shifting on the workstand, and it seems fine. But when you actually ride the bike uphill and downshift to the little chainring, the chain drags and rattles against the cage without ever dropping off the middle ring.) Here's your fix: Pull the crank so the derailleur is free from the chainrings. Grasp the outer side of the front of the derailleur cage with pliers. Bend the outer front of the cage inward. Bend from a point about 1 inch down from the front tip of the cage. Bend just a tiny bit -- a millimeter or two. That configures the cage to "kick the chain off" the bigger ring more quickly. Now readjust the limit screws and cable tension. If it's still a bit sluggish, bend the inner plate slightly inward at a point about 1/2 of the way between front and back. That makes upshifts (transfers to a bigger ring) easier.