The first sign is a very slow degradation of performance and an increase in friction. You'll not notice this over time, its only when fitting a new chain that the difference is clear.
Bad and slow shifts slowly increase, as does chain noise.
You can either run the whole transmission system in to the ground, and replace cassette/chain/ 2xjockey wheels and the chainrings, or you can change the bits that are worn while getting more mileage out of the chainring.
Be advised that chainrings are stupidly expensive, and a cheap big single chainring costs more than a cheap cassette.
The Jockey wheels are surprisingly important - if yours wobble side to side, consider changing them out too (they're relatively cheap)
Separate matter - if you have doubts about your LBS and their advice, just ask another one. Nothing wrong with a second opinion.
Some people just want their bikes to work (top efficiency is not that important) and other people want best efficiency at all times even if it costs more.