Practice, practice and more practice! There really are no magic tricks to getting better at anything. Repitition is really the key. Of course, you want to make sure that your form is correct. Focus mitt and heavy bag training are good tools.
Talk with your instructor as well. I'm sure they'd be able to help you out.
First advice to every beginner... Are you tensing up first, gathering strength for a monsterous push of the fist? Stop that. Loosen up. Clenching muscles means you have to unclench to move. It slows you down.
Be loose and fast. Strike with no pre-clench. Even if you chamber a strike before lashing out, don't be clenched in that position. You only need all your muscles to tighten at the instant of contact.
Same thing for breaking boards. Speed is the thing. How fast is your hand or foot going at the moment of impact. Add power only when your structure and the target are positively engaged. You cannot power your way through air. There is no resistance. You are wasting strengh muscle-against-muscle prior to actual contact.
Your own instructor will surely report you are tensing up. He'll be able to see it in the set of your shoulders, etc. If he says "relax" that's what he means. Don't clench. Be loose. Be fast and only add power when there is something to power against.
Throwing your weight behind any move means you will have first made an effort to gather your weight. That effort is costing you time, slowing you down. You can't throw your weight any faster than gravity allows it to fall. So it is a waste.
But if your foot work is such that when comes the instant of contact you are counteracting against the whole earth, that can be a whole lot more than your weight. It can be as much as you can lift at that angle.
But first it must get there. And it has to get there fast. Getting there late is the same as not getting there at all. You will feel it when it is right. Breaking boards and stuff will be easy. It is all just speed and timing.
Firstly, I would have to agree with the others in saying relax. A relaxed stance will give you speed.
Power comes from your feet. So for good power you need to have good balanced footwork. There is an old Chinese saying, power is developed in the feet, governed by the waist, and expelled through the hands. So, to find your power you need to see punching as a full body motion not simply one of the arm.
One thing the instructors don't want you to do is to "push" the bag. That means you need to "snap" your punch. The same with the backfist when you learn it, don't push, snap it. Speed is the key. Try to remember to hit with only your first two top knuckles (try to make sure that your thumb is under the middle knuckles - look at an ATA patch to see the placement). If you see that all of your knuckles are red, you are punching wrong.
Practice snapping a punch/correct punching - bring punching fist to bottom of your ribs at your side. Your hand is clenched into a fist but palm is facing up. Move fist forward, sliding arm along the ribs. As soon as your elbow is at your side, turn your fist over with the palm down and snap that punch - BAM! If you have a heavy dobok (most of us beginners do not), you will hear your dobok top "pop" when you punch hard and correctly.
I realize when you are doing punching drills, you're probably in a fighting stance but the snapping is what you're looking for. Your strength will build.
Make sure your wrist is straight when you connect! If you roll your wrist when you connect, it will hurt BAD for at least a month!! Doing push ups on your knuckles can help you develop that muscle memory.
I'm a traditionalist so I believe makiwara work is ESSENTIAL to developing a powerful punch. If you're an adult with no history of wrist or hand injuries, I suggest seeking out expert instruction on constructing and using one. As part of my last belt test, I demonstrated a three board break with a reverse punch. I truly believe this feat would not have been possible without the initially frustrating friendship I have with my makiwara.
I find that punching with weights helped me quite a bit, start with light weights and then gradually go heavy, if you can punch (with proper form mind, I can't put enough stress on the fact that punching with weights doesn't mean whole lot if you don't practice with proper form) with weights then punching without becomes easier and easier. There are other weight exercises that assist in developing punching power, most exercises involving shoulders, triceps, chest or core are great.
If you want to stay purely traditional and not bother with weights, push ups on the knuckles are another great way to develop punching power. And as already stated proper technique, and relaxation are key.