Grip shift ftw! They break FAR less often then trigger shifters. I work in a shop, trust me.
Anyways it could be a number of things. How old is the chain cassette? Have somebody put a checker on the chain so see how worn it is. If it is really worn out replace both the chain and cassette, if you don't it will create more problems. If it is within the limits of wear it could be the cassette that is worn.
Has the deraileur been adjusted since the shifter was put on? I'm assuming that they replaced the cable when this was done. if you don't know how to adjust them i wouldn't start turning screws because thats not the problem at all. To adjust tension, turn either the barrel adjuster on the deraileur or on the shifter.
It is a possibility that a link is frozen or bent but considering everything else that would be the last thing I would check.
From what you describe it really sounds like a deraileur adjustment so I would start there.