2004 Corolla Engine Cutting Out

Greetings all-

First time poster...help and patience appreciated.

I recently purchased a 2004 Toyota Corolla S, manual transmission, with 33k miles. At random times, the engine will simply shut off. It seems that this is more likely to occur on the first drive of the day. Sometimes it will happen on the highway, other times, at low speeds. Still other times, it doesn't act up at all. When I see it happen, I am usually able to downshift and pump the accelerator to get it going again before it completely dies. However, when it does totally die, it then takes 6-7 tries to get it started again (it will fire but won't remain running). To me, this sounds like a fuel problem, perhaps a fuel pump going bad. Any other ideas?

Update:

The mechanic claims that this may be a fuel problem, and told me to run a few tanks of premium gas through it and see what happens. I bought the car with a full tank of gas, and I'm pretty sure it was not premium. It has been fitted with an aftermarket (I believe) TRD supercharger, and the mechanic said that these superchargers are quite picky on the type of fuel used.

More information on the problem...I've been running it about two weeks now on premium gas, and the problem is not getting any better. The most problems occur during the first drive of the day after the car has sat out all night. The colder the day is, the worse the problem is. It seems the engine doesn't completely shut off so much as it just begins missing out (though this will progress to a complete "die" once in a while). When I drive the car to work in the evening, it never sputters...any help?

if it does it on cold damp mornings or when raining probably bad plug wires. open the hood at night in the darkest place you can find and see if you see any sparks