Manifold pressure isn't actually pressure, it's a measure of vacuum in the induction system. The more air each cylinder gulps in, the higher the vacuum present, the higher the manifold pressure. It's a measure of how much air the engine is sucking down. The simplest way to think about RPM, MAN, and Mix is this...
Mixture = Gas
Manifold pressure/throttle = Air
RPM = Engine speed
Prop pitch is based on what revs you set. Mixture is used to set the fuel/air mix inside each cylinder. Throttle controls the amount of air that enters the cylinders. If you leave the throttle wide open, you can actually control the power of any engine by just using the mixture control. Though you better be careful and not detonate the thing.
As manifold pressure rises the engine is torching off more fuel and air per cylinder. Most engines max out at barrometric pressure +- 3" simply because the vacuum inside the intake manifold matches outside air pressure. To get more bang for your buck you need a turbocharger or a supercharger. Either one will compress the air inside the intake manifold, making the pressure increase up to (and above) twice the ambient air pressure. This means you're getting a LOT more bang in every cylinder, and creating a ton of power in the process. The more power you create, the more force is exerted by the prop on the air, and the quicker you move.
About prop pitch: it's automatically controlled by a flyweight-driven governor. If you set 2500 RPM, the governor starts screwing with either oil pressure or an electrical motor to park the engine at the number you told it to. Lower revs mean a lot more torque goes to the prop, slightly reduced fuel flow, and a slower speed. But, the engine doesn't have to work all that hard to maintain lower revs; especially the big radials. In addition, lower revs increase the blade pitch. So while you're not moving the engine that fast, the prop is taking a much larger "bite" out of the air. Combined with reduced manifold pressure and leaned-out mixture, this is how WW2 aircraft were flying 8-10 hour missions. For high-speed flight you crank the revs up to max (usually around 27-2800) to flatten out the blade pitch. Now you're not letting the prop take as big of a "bite", but you're moving more air because the prop is swinging a LOT faster.
Think of it this way. You can either take big, slow, steady bites of that pizza. Or you can have a cheerleader on crack chow down on it. It's entirely pilot preference. To go fast you set high RPM, which flattens the prop blades out but drastically increases the speed the prop is spinning at. As a result, you move REALLY fast, but can't go all that far because you're torching off a gallon of gas every 10 seconds. Now to really haul bones, you need to set the mixture at best power for that altitude. Leaving it leaned out while suddenly flooring the throttle and RPM means you'll move a little faster, but not much. Setting RPM, mix, and throttle for various conditions does seem like playing piano. Once you learn the basics, you know where to set all three to get the airplane to do what you want.
John Deakin said all this a lot better than I ever could, so here's some links to snore over.

If you read a few of his other articles you'll learn most of the important stuff about where to set RPM, mix, and manifold pressure to get the best result.
Manifold pressure sucks!Those marvelous propsMixture magicPutting it all together-----------------------
Flakbait [Delta6]
Delta Six's Flight SchoolPut the P-61B in Aces High