I'm not sure how you would check to see if your tranny is slipping without going to a repair shop. It's more of a "feel" thing. If engine RPM is increasing steadily and your car isn't moving in relation to the RPM's then this would lead to transmission slippage (based on everything with the engine looking like it checks out). Chances are your transmission fluid will be a brown color and smell burnt. You could have bad bands or your torque converter could be shot.
