Do cassettes and chains need to be replaced at the same time?
You may need to change your cassette with your chain due to damage, but only if your bike has missed out on some previous maintenance. If you ride one chain beyond the point of wear, it will damage both cassette and chain rings to different degrees.

Your chain is a consumable component on your bike. 

For most people, 1200-1500 kilometers on a chain is a safe distance to ride it before it stretches enough to damage the cassette or chain rings.

If you treat changing your chain like changing the oil in your car, and just do it based on mileage, regardless of the actual wear on the chain, then you can continue to use the same cassette through (on average) 5 chain swaps.

The chain rings, at least a good quality set, are thicker metal, with a larger diameter, so they don't wear as quickly.  Usually, I get 10 chain swaps before I need to change the chain rings.  But they will eventually need to be changed as well.

I hope that helps.