Well, first I must say that Fuji makes a great bike. Although the key to a great bike is having a shop that is experienced enough to set it up CORRECTLY! It sounds to me that someone may have been a bit inexperienced setting up your bike... in 17 years as a mechanic, I've never seen a cable fray because it was clamped too tight. Did the cable have a cable end on it?(metal thing covering the end) Although it can happen.... my point to all of this is that it sounds like this bike was a learning experience for someone, and now unfortunately for you! I would take it back to Performance and have their ace mechanic go through the entire bike... if that isn't possible then I would take it to another LBS. They should be looking for a bent chain, derailleur hanger, derailleurs set up correctly,and the fact that the cables are fed through the shifters correctly and also run through housing that isn't frayed or has wires sticking from it and simply going through the entire bike.
AS for shifting on hills, what I normally try to tell people is try to quess what gear you need to be in before you start. If you happen to shift during the hill and want to shift to your small ring, then shift up one in the back first. This takes up more slack for your front derailleur and generally helps quite a bit. Remember that when you are shifting during the hill to be patient, and try and put less stress on the pedals during the shift... once the chain has moved then you should be all set.
I hope someone takes their time to simply check over your bike; it's simply unacceptable for someone not to enjoy their new bike! Have fun and keep us posted!!