My engine starts and then it dies immediately after that. What would cause that to happen?

It is a fuel management or igniton system issue.
The first things to check are the Idle Air Control Valve, IACV, and the Fast Idle Thermo, FIT, and possibly the coolant temperature sensor.

The IACV controls idle under different loads but can affect start up if it has failed or is failing. It is located on the intake manifold in the back near the throttle body. The FIT acts like a choke and is located under the throttle body and usually bolted into it.
They both have coolant lines running trough tem, so check for signs of leakage.
Each also has an elecrtical connector, the FIT may not. Disconnect one at a time and see if that has any affect.
The coolant temperature sensor is near distributor and can be confused with gauge temperature sensor. If you get into that far, you may want to get a Hayne's manual from a parts store or I can probably give you a diagram of where it is.
You could have a generall issue with the intake being dirty.
I would first check the fuel pressure though. It should be 45-50PSI. You can check it maually witha gauge at the fuel filter test port. It should come up as soon as you turn the key to the ON position.
Check that and remove the thottle body to see how dirty the intake is. Cleaning the IACV, FIT and TB may help a lot but make sure you clean them form the opposite direction of air flow so you don't clog it worse.
Let me know how it turns out and we can go from there.