My preferred method is to start from a fresh install. I inspect, generally replace the cable and housing unless they look and feel like new, the cable and housing for corrosion and also for tight bends which can restrict the smooth movement of the cable in the housing resulting in sticky shifting.
I disconnect the cable from the pinch bolt on the front derailleur then shift onto the small chain ring, which occurs when disconnecting the cable, and the easiest gear on the cassette/freewheel. Using the trim screw I adjust the front der. until there is 1-2 mm of distance between the chain and inner place. I check the front derailleur adjuster, if it has one and may be placed between the housing , making sure it is in a near closed position, meaning turned almost all the way in for the least amount of cable pull/tension... I always leave several turns from the bottom for tension adjustment increase or decrease tension. I check, by holding the cable in my hand with some tension, the shifter position by pulling on the cable and shifting up and down and leaving it in the least position...small chain ring position. Connect the cable to the pinch bolt making sure it is in the correct location on the der. and the tension, you are pulling the cable taut but not crazy tight. Tighten the pinch bolt. Shift from the easiest cassette gear a couple and shift onto the big chain ring. Shift down to the hardest gear and check the spacing between the chain and outer der. plate...it should also be around 1-2 mm. Shift up and down the rear derailleur checking for der. rub. Shift to a rear gear at the mid point of the cassette and aggressively shift up and down the front derailleur checking for proper spacing and that the chain doesn't derail. Shift onto the small chain ring and easiest cassette gear checking for proper spacing. If good you are all set. If it needs a bit of adjusting you can use the barrel adjuster to increase/decrease the cable tension until it is correct.
Good luck and keep us informed.