One way to tell if a chain is worn is to simply use a ruler. The distance between 24 pins (12 links) should be exactly 12 inches. If it's 12 & 1/16 - the chain should be replaced. If it's 12 & 1/8, there's a good chance your cassette will need to be replaced too.

So if you do replace your chain and suddenly it starts "skipping", that's a sure sign that your cassette is worn too.

As far as replacement chains go, I like the SRAM chains with a "PowerLink" connector. A PowerLink allows you to take your chain off and put it back on without using a chain tool.