The easiest way is to use a chain length checking tool, there are many types on the market. I have used the Park Tool CC-3 Chain Wear Indicator

To check the chain hook the left hand side onto the chain, as shown below (in this picture the 0.75% side has fallen in):

If the other side falls into the chain then the chain is worn. If the 0.75% side falls in but not the 1.0% then the chain is slightly worn and should be changed soon, if the 1.0% side falls in then replace immediately.

This is what the manufacturer of the tools says:

A worn chain shifts poorly and wears sprockets at an accelerated rate. The CC-3 is a "go, no go" gauge designed to accurately indicate when a chain reaches .75% and 1% wear, the points at which most chain manufacturers suggest replacement. Made from precision, laser cut steel.

As for changing chainrings, I would only replace if:

1. The chain is skipping because of wear of the teeth
2. You get chainsuck even when the chain is clean (small chainrings are more prone to chainsuck)

For cassettes I would not consider changing unless it skips when you have put on a new chain.

The best tactic is to change your chain often enough so that you minimise the wear in your cassette and chainrings.

Johan Bornman (www.yellowsaddle.co.za) makes the point that commercial chain checkers as shown above are not accurate (read more here) and suggests using a steel ruler. You measure from here:

to here (chain shown below is not worn)

If the chain is worn more than 1/16th of an inch (as shown below) then replace