To enter the climb, it's power, attitude, trim. That is, start by setting climb power, as you already know. Remember to compensate for adverse yaw with the rudder. Immediately raise the nose to the climb attitude. Your instructor will teach you to recognise what this looks like in your aircraft: e.g. there might be a part of the dash that lines up with the horizon. Your airspeed will drop away to the desired airspeed for the climb: usually the best-climb speed. Once you're stable and balanced, at the correct airspeed, use the elevator trim to remove the need for pressure on the stick (or yoke).
To level off from the climb, it's attitude, power, trim. Start by lowering the nose to a level attitude. You need to start doing this before you reach the desired height, as it takes a few seconds to level off. Once you're level, the airplane will speed up. When you reach cruise speed, reduce the power to cruise power, and compensate for adverse yaw. Again, once you're stable at the correct speed, trim away any constant pressure on the stick.
A gliding or powered descent is similar. Power, attitude, trim to enter the descent. First set the reduced or idle power, and wait for your airspeed to decay to the correct airspeed for your descent. When you reach this airspeed, lower the nose to the glide attitude. Once the attitude and speed are stable, trim the aircraft.
To level off from the descent, it's power, attitude, trim again. Anticipate reaching the height you want, just like when climbing. Raise the throttle to cruise power. Adding power will naturally cause a pitch-up tendency in most airplanes, so let the nose come up to the level attitude, and then you'll need forward pressure on the stick to keep it there. Once you're stable and level, trim away that forward pressure.
So for most of these attitude changes, the sequence of events is power, attitude, trim (PAT as a mnemonic). Leaving a climb is different: there it's APT, because lowering the nose first lets the speed increase to the cruising speed before you need to touch the throttle. Whenever you change power, you'll need to use rudder to compensate for adverse yaw.
Generally, you maintain the correct rate of climb or descent by keeping the correct airspeed, not by looking at the vertical speed indicator. Usually you just want the best rate of climb or descent, and you'll already know what airspeed achieves this, so it's easy to just check the airspeed indicator every so often and pitch up if you're too fast, or down if you're too slow. There are circumstances where you need to keep a particular rate of climb or descent, but that's not the norm so it's not what's taught first to student pilots.
The Pilots' Operating Handbook for the aircraft you're flying will tell you when carb heat is needed. In general, you should have it hot when you're at idle or low power (i.e. in the descent), and cold the rest of the time.