4Runner engine trouble

I own an '89 4Runner. It has a problem of dying all of the sudden. The problem baffles me because it’s like a “ghost” car and it only dies a handful of times. The battery connections are new, and I gave it a recent tune-up, so it starts just fine. I drive it a few miles and then I come to a stop. Everything is still fine. However, when I start applying the gas to get moving again, the engine cuts out entirely and it takes several tries to get it started again. Oh yeah, a mechanic also checked up with a computer of the power between the battery and the alternator and nothing was wrong. If anyone has any ideas or anything, that would be great. Thanks.

My first suspect would be a failing fuel pump.

First, change the fuel filter. Then, check the fuel pressure. Then, pull a spark plug wire and check for nice, strong, sparks during cranking.