What could be the problem with a 1999 Plymouth Breeze if when turn the key the engine doesn't start it just makes a clicking sound like the sound a gas grill makes when trying to light?

sounds very much like a dead/weak battery, if the dash lights/dome light/radio/etc dim out when you try the key that's likely it.

My lights turn on fine, put in a good battery and still have the same problem, any idea what the problem could be?

Answer

Starter (solenoid)

You should check your battery connections. Make sure there tight