My chainsaw will start but it boggs down and shuts down when I pull the trigger to rev the engine..

Engine may be too worn or may have run out of oil at some point, compromising the bearings.

Will the engine start without the pump connected?
If it will reconnect the pump, supply the pump with water, pull the trigger and try to crank the engine. If it starts, let go of the trigger and see if the pump boggs the engine down. If it does, the pump is defective. Return it to Sears for another new pump.

That could be a number of things, clogged fuel filter, faulty ignition coil, battery voltage low due to old faulty battery/bad alternator. Recommend finding a reputable mechanic that will properly diagnose the problem.

remove the muffler and try it if ok open the muffler and clean out the carbon from the screen

Its running too lean, this could be because the fuel filter in the tank id blocked, or there is dirt in the carb.

Maybe this will help.

The choke system is connected to the half throttle position, if you lift the throttle trigger the choke pull will automatically return, whilst trying to start do not lift the throttle trigger, the starting procedure is, pump the primer bulb as many times as it takes to 3/4 fill the bulb, pull the choke pull fully out, this will switch the saw on and engage the half throttle, crank the engine until it fires once, do not continue to pull,it will actually not run on full choke, if you do continue to crank with the choke fully out, you will very quickly flood the engine, now push the choke fully in, now crank the engine until it fires up, lift the throttle trigger once, this will dissengage the half throttle, and it should now come down to idle.

Check the fuel filter for plugging as well as the air cleaner and muffler. Make sure the choke control actually opens the butterfly plate in the carburetor when the choke is pushed off. Check the quality of the spark at the plug by laying the removed plug on engine metal (wire on plug) and pull through. The spark should be blue and snappy. Plug gap is 0.020". The insulator should be medium to light tan when the engine runs correctly. If still having problems, check the lower chamber diaphragm in the carburetor for being hard or cracked as it directly controls fuel flow through the carburetor. Hope this helps!

hey pete
This is easy, prime bulb 5 to 7 times with choke lever full out. Pull string, once engine starts turn off choke and pull string again. ( throughout process do not pull trigger )Engine should start, let it idle 30 seconds, then finally pull trigger to rev up engine.

May need to overhaul the carb. I suggest that you replace the fuel line and fuel filter too.