I turn the ignition on and wait for the lights to go out. Including the engine light. When I start it, the engine light comes back on for about ten seconds. After that, it goes out and it never comes back on. It just started doing this. It does it when it's hot or cold. It's an 08 EGC, 4800 miles. Do I forget about it or have it checked out?

 

That is the indication that you have a logged diagnostic code. Get your manual out and check it.
Do a search. Plenty of threads on here that will tell you how to do this.