Rpm shoots up to 3500 dwn to 1500 up again down again .

Check for vacume leak including intake. Spray some carb cleaner around intake and vacume hoses and listen for engine rpm to change.

First clean the fast idling valve, if that does not fix it, second there may be a vacuum leak

TRY REPLACING YOUR PCV VALVE

That would be your engine "choke" functioning. It's not a problem. The choke engages when engine is cold and shoots extra fuel to the cylinders to idle the engine up especially in cold with. Once the engine has sufficiently warmed up, the engine will idle back down.

It may be caused by a vacuum leak, make sure to check all your hose for cracks and/or hoses that may have popped off

it will do that, it is normal, should go between about 600 and 1500 rpm...
----------- thats just a normal warm up cycle, it will change the engine speed and mixture as combustion gets better with the heat.

could be your clutch is not fully disengauging or on its last leggs check if you can ajust the freeplay at the clutch cylinder thats if your car is a manual

I am having the same issue after I changed my battery today. Upon doing some research it seems like there is some way of aiding the computer with proper idling. I haven't done it yet... plan on going to the dealership where I got the battery, but from my understanding you disconnect the negative line from the battery for a few mins, reconnect, and then apply gas at 3000 rpm for a few mins until the fan kicks on. AND this will help reset your car to proper idling. Good luck

I had the same problem with my honda. On the throttle body there is a screw that controls idle speed. If your idle speed is around 800-1000 rpm in neutral, tighten the screw a quarter turn each try until the idle speed is around 1500 RPM. Did this stop the idle?

It may be the throttle position sensor (tps)