My wife recently bought a Schwinn Ranger at Target. It started off missing and having difficulty with shifting gears. It became apparent that the problem was in the front derailleur. I watched a video on how to adjust high and low limits. Problem is, when I get one setting okay, the other gets bad. As I write this, I can get 14 out of 21 gears okay; it won't shift to the large front sprocket.
Two notes: I have no means to elevate and shift while tuning and I have to tighten up the cable, take it for a ride, see what it's doing and then go back. Note two: I see movement when I adjust the low limit screw, but little,if any movement when I adjust the high limit screw. I've spent most of the weekend on this, with no result. Help!
First of all, it would be very helpful if you can run through the gears while you're working on it. Maybe your wife or someone else can hold the back end of the bike up while you spin the pedals. To begin, make sure the bike is in the lowest gear... big cog in back and small chainring in front. Make sure the grip shifter is set at "1" for the front derailleur. Loosen the bolt on the front derailleur that holds the shifter cable, take out any slack and re-tighten the bolt. Now on the derailleur itself, locate the "low" limit screw. Turn the screw until it just touches the chain, then back it off a hair. Shift to the middle chainring and run through the rear gears, when you get it to the smallest cog leave it there. Now shift the front derailleur to "3", the large chainring. If it won't move then you have the "high" limit screw too tight. Back the screw off and keep trying to shift until it goes to the large chain ring.
Then do the same as the "low" limit. Adjust the screw 'til the chain just touches, then back off a wee bit. Now shift the front to "2" and run through the rear gears again, making adjustments at the barrel, on the grip shifter itself, to tweak the front derailleur to eliminate any rubbing. Take a look at Alex's tutorial again too for a refresher. Good luck, and be patient, it can take time to get the adjustments just right.
Just a quick suggestion for those without a workstand to elevate the bike - find a table with the right height so when you hook your seat on top, it lifts the rear wheel off the ground and gives enough space to turn the cranks. Poor man's workstand.