Road bike question!

My road bike has extremely sloppy rear shifting. No matter how well I set it up, it soon reverts back to needing to be constantly over-shifted in order to change gear. I’ve replaced the inner and outer cables, but it’s still pretty rubbish. Once in gear, it’s absolutely fine with no skipping or jumping, so I don’t think it’s transmission related.

By my reckoning it can only be:
Shifter
Mech
Cassette / Chain (outside chance)

(I can get it setup perfectly in the garage, but it simply doesn’t stay that way, so I’m confident that, in this particular case, it’s probably not user error 🙂 )

I’m probably at the point of replacing bits, but as I don’t have spares I’ll need to go and buy them, which means that I’d rather not buy what I don’t need. Has anyone got any suggestions on how I can tell which part is most likely to need swapping.

Recommendations, advice, free components very much appreciated from the ST hive mind.