When I start my 98 cutlass(115K), sometimes the starter engages and turns the engine for a couple seconds and then it sounds like maybe the starter loses contact with the flywheel and then it just spins without turning the motor.

The battery checks out okay.

So I am assuming something isn't right with the starter.

Does anyone have a link or a picture diagram as to how to remove the starter

Or can someone explain it to me for this car.

What looks like the 2 bolts for the starter are in a vertical position, but when I think of pulling a starter, I would think they would be positioned horizontally.

I'm just looking for some advice before I risk opening a "can of worms"

take the two bolts out that are verticle. sounds like you have worn flywheel teeth

1 - Jack up car;
2 - starter is under front near the exhaust manifold
3 - you need to disconnect the wires connecting the solenoid to the starter, and
4 -As said above, there are two bolts (a long one and a short one).

However, it does sound like you have a dead spot on your flywheel. When it starts to spin, does it start?

yeah, it has been starting if you keep trying it, but it may take 5 or 6 tries before it turns the motor enough to start it.

It probably turns until it hits a good spot. If it turns over, probably not the starter, unless it clicke 3 - 4 times and then on the 5th time it clicks and spins, then it may be the starter.