re: OP yes cassette is worn. you tend to use certain sprockets more often than others so they will be the ones that will jump when a new chaing is fitted.
buy a chain wear tool, if you don't already own one, and replace at the .75 % wear limit; that way you can get aoubt 3 or 4 chains before the cassette is so worn that new chains jump.
ofcourse as you reach the end of you cassette's life, the worn teeth will rapidly wearing a new chain as the two mesh together.
it is hard to tell wear on the chain rings and the cassette; the chain rings need to be really badly worn before the cahin starts to skip and it is quite noticeable - you should be fine with yours for a while yet.
buy a chain wear tool, if you don't already own one, and replace at the .75 % wear limit; that way you can get aoubt 3 or 4 chains before the cassette is so worn that new chains jump.
ofcourse as you reach the end of you cassette's life, the worn teeth will rapidly wearing a new chain as the two mesh together.
it is hard to tell wear on the chain rings and the cassette; the chain rings need to be really badly worn before the cahin starts to skip and it is quite noticeable - you should be fine with yours for a while yet.