Car hesitates, and RPMs go up on automatic

I recently purchased a 2002 Isuzu Axiom, and have encountered an issue I need help on before I take it to a mechanic. While driving, though not everytime does this happen, only after an extended trip (like running errands) I come to a stop, and when it's time to go, I press on the gas, and it hesitates (in between the 1st and 2nd gear), and the RPMs go up to 3. If it sits for a little bit, it goes back to normal, until I make a few trips. And I am wondering if the revving RPMs, could of caused the water pump to go on me as well? I didn't mash my foot on gas when these episodes happen, but it makes me wonder with the engine revving up a little more than it's supposed to, could it of made the water pump break? Any help would be greatly appreciated, and if anyone could tell me where I could also find a dealer in my area (St. Petersburg, Fl) that has a computer for my car to look up to see if it has any codes or flashes on it, since Isuzu is no longer around... Thanks again for any help!!!
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