Chains need to be replaced periodically because the pins that connect the links wear down with use. As the pins wear, the chain elongates a little bit. That extra length doesn't mesh quite perfectly with the teeth on your cogs and wears them down. So in a nutshell, a worn chain also wears down your cassette or freewheel, not the other way around.

If you replace your chain at regular intervals, your freewheel or cassette will last a lot longer. Your local bike shop likely has chain wear gauges, which are a simple tool. You can also measure wear easily with a ruler since the pins on an unworn bike chain are exactly 1/2 inch apart. Measure a 12-inch section of chain, starting exactly at a pin. If the pin nearest the 12-inch mark is more than 1/16" away from the 12" mark, the chain is "stretched" and should be replaced.

Chainrings tend to wear more slowly because there are more teeth engaged with the chain at any given time to share the load. Do you need to replace yours? We'd need more info to help you determine that. If the teeth on a chainring show an uneven "shark fin" or "saw blade" profile, that's a visual clue that the chainring is worn.