Drain the system, refill with clean tap water, drain it again, refill with tap water. Add a bottle of 7 hour Prestone flush and do what the directions say, drive it for the full 7 hours. Drain the system and refill with water, do this about 3x. Then change your parts. Once you have the water pump off you will have full access to the areas around the cylinders and you can spray it out some more. Probably the most important part is TURN ON YOUR HEAT to hot so the flush gets in the heater core, yes you are going to sweat your a$$ off, more or less, depending on where you are. When the pump is off so will be one of the heater hoses. Disconnect the the other one and shoot water in first one hose then the other, do this till it runs clean [may take a while]. It is also a good time to replace the heater hoses with new ones, don't try to pull the hoses off the heater core bungs, remove the clamps and slice the hose lengthwise then gently peel it off.
It is also a good time to replace that heater bypass tube on the water pump. You can reuse the one thats on there but I'd recommend either replacing it with a new one and paint that with a high heat paint OR use brass nipples with a 90 degree elbow, that will last forever. I don't remember the thread off hand but if you take the new pump to a plumbing supply store they can fix you right up with if I remember correctly a 5" straight nipple, 3" straight nipple and a 90deg elbow. Also do use distilled water to make up your 50/50 mix of coolant and replace the pressure cap too. The tstat from the dealer has a hole in it, that hole goe in the 12 O'clock position, straight up, it helps get air out, alot of the aftermarket ones don't have this.