What does usually worn first, the chain or the cog? I have an old bicycle that I would like to repair. Basically the chain is always "clicking/jumping" when I pedal. 
I know this is either a worn chain or cog. My question is what worns first?

If it's the chain I might actually give it a try and switch it. If it's the cogs I don't think it's worth the money and I doubt I can switch it myself and I don't want to spend too much money with this.
Tricky one.

You will go through several chains before going through a cassette.

A worn chain can be measured quite easily with a gauge.



Measuring wear on a cassette is more difficult. It is usually the middle set of sprockets which wear first - due to their more frequent use. If examined carefully - you may notice the teeth on them thinning.