Very Hot Radiator

How does the temp do down the road?

Do this simple test:
While it's cold, with the cap removed, make sure it's full of coolant (just a bit above the tubes) and fire it up.
Check the radiator for either a steady stream of bubbles or an exhaust smell, if there is neither, then it's a good bet that there are no cracks or bad head gaskets.

You might also have your radiator flow tested, it's cheap and not much hassle, plus some places can do it in the car.
Do all of this, it's better than throwing good $$ after bad for a radiator, only to find out you have a bad head gasket or cracked cylinder head.

For the engine to be heating up that fast after just 10 minutes of idling, indicates either drastically reduced airflow or severe coolant flow issues.
I've had vehicles that idled for a half hour and you can still touch the radiator safely.

Oh, be sure you get a NON-ELECTRIC (manual) temp guage, if you decide to put one in the car.
They read more reliably.

Mark.