The front derailleur can be adjusted with the barrel on the shifter. The limit screws will only stop the chain coming off, and have no effect on the middle ring (assuming a triple ring up front). Bear in mind that the chain will rub if you select gear combos such as big-big (which you shouldn't by the way).
Does the rear derailleur jump in all of the cogs or just one or two? It could just need adjusting, hanger could be bent, could be a stiff chain link (unlikely I think), etc. etc. Almost certainly fixable though.
The park tool website has walkthrough for most little repairs, take a look on there.