I didn't see one or something similar in the catalog so here we are and I need help.

I had my chain skipping in the hardest gears on my road bike, 10speed cassette and double crank, which I don't normally ride in. But no issues otherwise
So I thought maybe my chain was to blame but never measures and don't have the tools for that anyways. I bought a new chain and now it starts to skip on the front sprocket.
Now I'm starting to wonder if I have worn out both the front large sprocket and the middle of the cassette to the point I just need to replace all of it... thats a little pricey and I'm cheap.
How should I proceed in trouble shooting this?

thanks in advance