At high throttle the engine puts out more power, but there is a corresponding increase in the load on the engine due to increased main rotor blade pitch, and so the engine stays at the same RPM's.) This overall increase in pitch that makes the heli climb is called collective control.
To get the heli to pitch forward or back, and to roll left and right, there are controls that are analogous to airplane elevators and ailerons. These controls are refered to as cyclic controls. The idea is to set up asymmetric lift on the rotor disk. (This is similar to what ailerons do to an airplane-one wing can be made to generate more lift than the other, and so the airplane rolls.) If there's asymmetric lift on the rotor disk, the plane of rotation of the rotor disk is going to change. For instance, the rotor disk (and the heli that is attached to it) might go a bit nose-down. In that case, the heli will transition out of a hover and start flying forward. Similarly, the heli can be made to lean back (nose-high), left, right, or any combination of these. The way this asymmetric lift is set up is to vary the pitch of each blade as it goes around. For instance, say you push forward on the cyclic control stick (the right one on your transmitter, which does the same thing as an aileron/elevator control stick on an airplane radio). This will make the blade pitch down as it travels through the forward-moving part of the rotor disk (usually the left side of the rotor disk), and it will make the blade pitch up as it travels through the backward-moving part of the rotor disk (usually the right side of the rotor disk).

What is gyroscopic precession?
This is a counter-intuitive aspect of helicopters, that even many advanced pilots don't clearly understand. In order to get the helicopter's rotor disk to tilt (for example) downward at the front, you increase the lift on the right side of the rotor disk and decrease the lift on the left side of the rotor disk. (This is assuming the standard clockwise main rotor rotation.) To see why this is so, consider the following example. If the heli is in a nose-down attitude, the forward moving blade travels downhill, and the aft-moving blade travels uphill. The blades travel level at the front and back. To get a hovering heli to go into a nose-down attitude, you need to encourage the forward-moving blade to start going downhill and the aft-moving blade to start going uphill. Hence, pushing the cyclic stick forward causes lift to be killed on the forward-moving (left) part of the rotor disk and increased on the aft-moving (right) part of the rotor disk.

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