Given that it is an old bicycle, it is likely that the chain or the cassette (or even chain ring) are worn down to the point that it's causing your symptoms. However, it's also possible that the derailleurs are mal-configured and your gears need to be indexed correctly. It may be worth checking this out too as it's a relatively quick fix.
As pointed out by OraNob, chains generally wear much quicker than cassettes and as a rule of thumb would be replaced more frequently. Chains stretch with use and if left too long can begin causing issues with your cassettes and front chain ring(s). As the teeth don't mesh with the chain properly, the result is a rapid wear of the teeth and will ultimately end up in both needing replacement.
Switching cassettes is not really much harder than changing your chain, you just need the right tools for the job. You can pick up relatively cheap cassettes from your local bike shop or online, so cost shouldn't really be an issue here. You need a chain whip to keep the cassette / cogs still, and another tool for unscrewing the cassette. These will depend on your make/model of cassette as there are differing standards.
Indexing bicycle gears
Changing a bicycle cassettte