Don’t be too quick to blame the computer. At startup, nearly all motors operate in “open loop” (read “normal”) mode with the bare minimum of controls/sensors active. It could still be the comp, but if it is, it is beyond most “average” DIY attempts. First, check out the things you can do something about.
Always check ignition first.
Forget the “hold the wire a short distance from ground” method. The only reliable DIY way is with an inductive timing light. If it flashes on all wires, the ignition is ok (timing is another matter). Wires, plugs, cap, rotor, coil/CD box, and controllers are the usual bad parts (in that order).
Based on the info you have posted, it really sounds like a bad fuel pump. To check: with the ignition in the ON position, find a Schrader valve (like on a tire)in the fuel line. There should be about 60-80 psi there. If you listen closely near the tank, you should also hear the pump running. If not, look for a bad fuse or tripped crash/inertia switch before replacing the pump itself. Advice: Don’t run the new pump low on fuel. It uses the fuel as coolant.
After that, you should fear a broken timing belt. $$$. (The manufacturer’s recommendation of 60k mi is a real upper limit. All the failed belts I have seen come in within a few thou of that number.) To check this: look into the valve cover (where you add oil). If anything (rocker arms) in there is moving when the engine is cranking, the belt is probably alright. If all you can see there is a heat sheild/baffle, look for a plastic plug on the upper, front face of the motor. The cam gear will be behind that plug. if it is turning, you’re probably ok. Finally, the distributor is nearly always run off the cam or timing belt, so if you have a spark, the belt is probably ok.
If all the above fails, you have reached the limit of normal driveway diagnosis. I pesonally recommend independent shops over chains or dealers, but take it to whoever you trust. Just don’t hand over your keys if you are even slightly uncomfortablle.