rough warm idle

What is your question?

When I start my car 141,000 miles in the morning or after sitting for a few hours it is fine but if I drive home from work about 30 minutes of driving, run in the house and change, or stop at the store, when I start the car again it dies. It has no problem initially starting it just idles at such a low rpm it dies. It I push on the gas after the car starts and rev it up to about 2 or 3 thousand rpm and keep it there for 10-30 seconds it's fine. Taking it to the dealership and thinking the diagnostic might be a waste of money if someone already knows what this is.