Moter runs fast then slows down

There are many things that could cause this problem,a misfiring cylinder ,low compression or idle speed not adjusted right,
Is this car fuel injected or does it have a carburetor?
if the timing is off, or the main problems i have seen are in the carburetor. does this car have an o2 sensor? a bad o2 can cause a bad fuel mixture and make engine load up at low idle speeds.
this car is a 1988 Camaro, what size engine , automatic or stick and is it carb or injection fuel?

has base idle speed and kurb idle speed. allows engine to close throttle opening when shutting down to stop pre-ignition or engine running on

If the check engine light is not on, its going to be tough to troubleshoot.
You would need to be able to duplicate the problem on demand to know if you fixed it or not.
Could be anything electrical, like a relay, or the coil, or ignition module, and so on.

It could be a relay that keeps the fuel pump running when the ignition switch is in the run position. My jeep had this problem, where the pump gets power in the start position but once the key returns to run, the engine shortly stops running because the pressure in the fuel line drops off as it is used by the running engine, it usually died within 3 seconds. my relay was under hood and there were several of the same type in there, I switched the fuel system one with the one for my AC(which only blows hot anyway) Hope this helps...

Check to see if there's power at the fuel pump when cranking the engine. If no power, check the relay and oil pressure sender switch. If the engine looses oil pressure, it stops the fuel pump from running to protect the engine. The relay is located on the firewall above the master cylinder.

Sounds like your fuel pump is dying.

You may have a vacuum leak somewhere in the brake booster. The vacuum is created by a tube leading to your intake manifold, and if you hit the breaks too hard, it'll **** the air out of the manifold in order to restore the vacuum- hence the engine dying. The leak could be anything from a visible hole to a pin sized hole.

If it runs at highway speeds then I doubt it's fuel filter. I'd check the operation of the idle speed motor. Here's a link to a diagram so you'll know what to look for & what to do. just scroll down to your engine (fig# 2 ) & click on small picture to expand it.
http://www.autozone.com/shopping/repairGuide.htm?pageId=0900c1528007f93f

Change/thoroughly clean the idle control solenoid. It may be sticking and causing rough idling.
Cleaning it is unlikely to be more than a short-term solution.