Welcome to C&V.
Presuming you want to get your bicycle back to being reliable without replacing it or spending a fortune, I suspect from what you've written that your chain is worn and your shifters or derailleurs may need adjustment. It is generally recommended that when changing a worn chain that you at least inspect and consider changing the chainrings (in the front) and the freewheel (in the back), as wear on these can accelerate wear on the new chain.
When it comes to wheels, I generally build my own, but I recognize this is not for everyone. Steel rims (which it looks like you have) are considerably heavier than alloy rims, and the brake pads generate less friction on steel wheel - especially when wet. If you are considering new wheels, look for something in the same size, with alloy rims.
No idea of where you are located, but Narhay may have a point. In some places, it may actually cost less to get an entire used bike (possibly even a nicer one) than it would be to fix what you have. This reality strikes me as a bit shameful and wasteful, but it isn't a reality I can change. If you have a local bicycle co-op, I suggest taking your bike there for a consult. Also, if you do decide to go with a different bike, consider donating your bike, as opposed to setting it out for trash.
Don't feel shy about asking questions. The folks here are quite helpful almost all the time.
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