1998 Oldsmobile Silhouette won't start - could be engine EL

First the obvious - as above - the problem is it won't start. I bought a jumper kit from an auto supply store, and hooked it up after I realized my batter was low or dead.

Without the jumper kit connected, I hear a click, but no rolling (or cranking, I think it's called).

With the jumper kit connected, I hear proper cranking. I think I can be pretty sure that the battery needs to be charged / replaced. However, I can tell the starter motor is cranking just fine when enough electricity is present.

As far as fuel delivery, I'm pretty sure I can hear the fuel pump motor spinning for a few seconds when I cup my hand around my ear and listen by the middle of the back seats in the van (fuel tank is right under where I'm listening, and pump is inside the tank from what I've read). This occurs when I turn the key to the position just before ignition.

I have searched several online forums and q&a, and have found that it could be the "engine electrical" or "engine computer." Is there any sort of quick fix or at least a test that would let me know if it is this before I take it to a repair shop?