The early Cherokee series (140/160/180/Warrior / Archer) are not fuel injected so the electric fuel pump does nothing except supply fuel to the carb - nothing you do with the pump after starting will have any effect on the engine. Having it on for takeoff is to ensure a positive supply to the carb in the event of an engine-driven pump failure.
With the primer, you need to pull it out to full length, pause (for the chamber to fill) and then push in. Next time you pre-flight, have a look at the engine. You will see a series of fuel lines going into the cylinder heads - these are the primer lines. Priming works because it delivers fuel to the intakes. Pumping the throttle will cause the accelerator pump to deliver extra fuel to the carb throat - which is under the engine. A few 'pumps' of the throttle and you will have an air intake box full of fuel - a great recipe for a engine fire.
These are all issues that should be addressed by your training organisation; some are not covered by the POH.