Think of it in terms of r's having to cancel.
Work = Force * distance = Torque * theta. If Torque = Force*radius then theta must equal distance(around the circumference)/radius so that the radii cancel.
Applied power = Force * momentum = Torque * angular velocity. If Torque = Force*radius then angular velocity = regular velocity / radius.
So now we have this system where we can see that some things are times r but other things are divided by r. How do we tell which is which? In general, the expressions that have masses in them end up having r multiplied. Momentum is mass times velocity, so angular momentum is regular momentum TIMES the radius; regular velocity has no mass, so angular velocity is regular velocity DIVIDED BY the radius. Force is mass times acceleration so angular force (torque) is regular force TIMES the radius; regular acceleration has no mass so angular acceleration is regular acceleration DIVIDED BY the radius.
Why is that the case? Hrumm... I guess I would explain it in terms of the way two properties transform from linear to rotational quantities. First: You convert a displacement to an angle by dividing by the radius. The car went 300 meters along the outside of the track? What was that in radians? Well, it was a 150-meter-radius track, so the car went 300m/150m = 2 radians around the track. Angular velocity and angular acceleration work the same way for the same reason: if I'm driving 30m/s around the track, that means that in one second I go 30 meters, or 0.2 radians, so my angular velocity is 0.2 radians/second---note that I still had to divide by the radius to get there.
So what about things with mass? It turns out that the "rotational mass" of a particle about a point is mr^2, where m is the particle's regular mass and r is the distance from the particle to that point. So when you form angular momentum by multiplying this rotational mass by the angular velocity, one of the r's in mr^2 cancels out the r from the angular velocity, and the other r sticks around to be a part of the formula.
Why is the "rotational mass" (more properly called the "moment of inertia") equal to mr^2? The derivation is that if a particle is moving with in a circle of radius r with a velocity v, then its kinetic energy will be (1/2)mv^2, but since v = wr because that's the definition of angular velocity, then the kinetic energy is (1/2)mr^2w^2, so if we think of I = mr^2 as its own interesting quantity, then the kinetic energy of a particle moving in a circle can be written as (1/2)Iw^2, which has the same basic form of (1/2)mv^2.
Is that good enough? We could derive things more rigorously if you'd prefer.