[askvictor]

As said, there are different schools of thought. The smaller cogs in particular will wear out the fastest; if you're using them a lot, the cog wears, which then causes the chain to wear faster, which will then cause the other cogs to wear faster.

[markandrewj]

It's sort of the opposite, a worn chain will wear on your cassette. Using a chain checker, or a ruler, you can measure how worn your chain is. If you replace the chain too late, the new chain will not mesh with the cassette, and it will slip. You can usually do this 3-4 times before the cassette will also need to be replaced. How long you will be able to use a chain is dependent on the riding environment, chain material, how well it is cleaned, and how well it is lubricated.

With an 8 speed bike, riding between 300km - 500km a week, I usually get around ~1000km on a chain riding outdoors, or about a month on a chain. Riding indoors on a trainer however, my chain almost never needs to be replaced.

For an 8 speed bike you replace the chain at 0.75 wear.

Depending on the bike you may choose to let both wear, and replace both at the same time. Eventually you will find the shifting sloppy, and the chain slipping however. This observation is from personal experience.

I find the idea of using string interesting, more commonly on the market you would see belt driven designs like the pinion.