Use your feet
The choice of where you place your feet is the major determiner of where you can go next.
Lead with your feet
Your reach extends only as far as you are tall, so to reach farther you have to move your feet up. Obvious, but new climbers tend to focus only on their hands and the direction they are going -- up -- and forget that their feet have to move up the wall too. If you do this, you'll become extremely stretched out on the wall. It's really hard to move out of this position. It'll be hard to see where to place your feet because your torso is pulled in so close to the wall, and it will be hard to actually make the next move. A better technique is to move the feet up, which moves the torso up, which increases the range of holds that the hands can reach. Moving the feet up first also bends the knees and hips which gives the climber leverage, and makes climbing up easier. You do want to stay in close to the wall, but as I'll talk about later, generally you want to twist and keep one side (not your front) in to the wall so that the other side is free to move. Avoid becoming too stretched out as it limits your options.
Place your feet quickly
When climbing, you want to save your strength for big moves that really require it. Try to avoid whittling away your reserves by using your arms when you could be using your feet throughout the climb. The muscles in your legs are larger and stronger than the muscles in your arms, and you want to try to use this fact to your advantage as much as possible. To do this, keep your feet on the wall! Avoid leaving limbs dangling. Before making a move, think about where your feet are going next. After you make the move, place your feet back on the wall as quickly as possible to minimize the time that you are holding your body weight with your arms.
Use your legs
As noted above, your legs are stronger than your arms and you should use this to your advantage. Your legs carry your weight every day, but routinely your arms only carry a fraction of it. New climbers tend to focus on their arms and pull themselves up the wall. One of the most common excuses I hear from people who think they can't climb is "but I have no upper body strength". This statement is misguided because massive upper body strength isn't necessary at all. Overusing your arms and neglecting your legs is simply bad technique. Practice using your legs to push yourself up, and using your arms only for balance. Step up to a foot hold and stand up on it by straightening your leg and keeping your hips in to the wall, while at the same time putting as little weight as possible into your hands. (This is easiest to practice on slab or on a wall that's a few degrees from vertical.) This technique will teach you to use your body mechanics efficiently.
Keeping weight out of your hands will also aid in not overgripping (holding on with more strength than is strictly necessary).
Pushing with your legs will also be covered in the section on stemming when talking about using opposing forces.
Place your feet carefully and firmly
Your feet should not make noise when you place them on the hold. If they are, you are wasting energy by bringing them down faster and harder than is necessary. If your feet are making noise then your foot placement is also likely to be a little sloppy and uncontrolled.
Climb using the edges of your feet. Aim to place the ball of your big toe on the hold. Avoid using the ball of your foot, as using your toe will give you a little more reach, and a little more leverage -- often that's all that is needed to get through the crux of a climb. Climbing shoes are designed to make this efficient. They draw your toes together, giving you a smaller surface to place on the rock, minimizing the size of the foothold you'll need.
Once you have placed your foot, press it firmly downwards and into the wall. Keeping tension in your leg will reduce the chance of your foot slipping, and will of course move you upwards.
See also the section on shifting your weight for advanced foot placement techniques.
Other foot placement techniques involve using your heel hooked around a hold to pull yourself towards it. I'll talk about these in detail in later sections.
Smearing is the technique of pushing the flat of your foot on the wall where there is no foothold. The friction between the sole of your shoe and the wall is enough to hold your weight up. It creates a point of contact with the wall to maintain balance while you make a move. As soon as you are able, move the smearing foot back to a foothold. The keys to smearing are the force and direction in which you push into the wall. Friction is proportional to the amount of force applied, so use a lot of muscle to press your foot into the wall. The harder you push, the less likely your foot is to slip off. Also you want to be going up (presumably), so the force should be directed slightly downwards (not just perpendicular to the wall), to give you a bit of help.
Trust your feet
Yes, the hold is small. Yes, you can stand on it. Yes, it will hold your weight.
Like a ballet shoe, climbing shoes are designed to cram your toes together into a point. This focuses your weight into a smaller area, making it easier to stand on smaller things. It's not natural to stand with only the ball of your big toe on a 5mm (or smaller!) wide ledge, but it is possible, and it is good climbing technique. Learning to trust your feet is necessary to climb harder routes strictly from a physical perspective. It is absolutely essential from a mental perspective to allow you to be comfortable on the rock. If you are constantly questioning if the hold is good enough to step on, then you can't concentrate on the sequence of moves you are executing, and you'll psych yourself out.
Understand gravity and the forces you exert to oppose it
It takes time, but after enough practice you will begin to develop an intuitive sense for how shifting your limbs changes your centre of gravity and how your body will shift on the rock. With that understanding, you can start using the way your body shifts to your advantage.
Keep your hips close to the wall
Keeping your body close to the wall accomplishes three things: first, it decreases the distance you have to move to get to the next hold, second, it increases the effectiveness of your footholds, and finally it decreases the amount of force in your arms that you need to hold yourself up. This is true whether the wall is vertical or overhung.
Picture the following: you are standing on two footholds which are at the same height, and you have your left hand on a generous handhold. The next hold is farther left. Imagine matching on the rightmost hand hold, then reaching for the next hold with your left hand, all while keeping your feet where they started. If the second hold is far enough to the left, this will shift your centre of gravity far enough to the left that your right side will start to spin out from the wall, and it can be very hard to keep holding on to the new hold with your left hand.
Now imagine the same initial situation and again match hands on the first hand hold. This time though, as you move your left hand out to the next hold to the left, also move your right leg out to the right. This will keep your centre of gravity over your left leg, and will reduce torque. As you do this, also twist your torso to the right so that you are reaching backwards with your left hand for the hold out to the left. This will maximize your reach (see the section on reaching backwards) and will make flagging with your right leg easier as it will be moving out in front of you, not out to the side.
A slightly more awkward, but possible technique is to flag with your left foot to the right. From the same starting situation, switch feet so your right foot is where your left was originally. As you reach for the hold to the left with your left hand, move your left leg to the right behind your right leg to counterbalance. This configuration may be useful depending on the rest of the climb.
When the points of contact between your body and the wall are all to the left or all to the right, gravity will want to pull your body out from the wall. This is called "barn-dooring", as your points of contact act like a hinge and your body rotates around it, away from the wall. To prevent this from happening, either make sure that your center of gravity is between your holds (for example, using your left hand and your right foot as opposed to left hand and left foot), or you can flag to balance out where you body is as compared to the holds.
Use opposing forces (Stemming)
We've already talked about the importance of pushing your feet into the wall when you climb. Stemming is the art of managing these forces to help you climb up. Think about climbing up the inside of a narrow chimney. You can press a foot into the wall in front of you, the other into the wall behind you, and use these forces to hold your weight up. Then, press your hands into opposite walls, and one at a time, shift both feet up a foot or so, then stand up on your legs. Repeat. This is technique of using opposing forces is called stemming, and it's applicable to many more frequently encountered situations.
Stemming is not useful only in chimneys or in situations where two walls meet at an angle. You will find it useful, especially when climbing a route with many slopers, to push off a lower hold to be able to reach for a higher hold.
Shift your weight
Often times, a hold may only be good if pulled at from one direction; however, you may not be pulling from that direction when you first use the hold. In order to get a better grip, you will need to shift your weight, so that gravity is pulling you into the hold instead of away. A similar problem occurs when making a horizontal or diagonal move, where reaching for the hold will move you off balance. To compensate, first shift your weight towards where the hold is, so that you remain supported while reaching.
Maximize your reach
People who start climbing want to face towards their next hold as they are reaching for it. Due to the way the shoulder is constructed, turning away from the next hold and reaching backwards for it actually allows a farther reach than reaching forwards does. Think about how you would reach for something that is far under a bed, and use the same twisting motion in your climbing.
Another tendency of beginning climbers is to not take full advantage of the holds available to them. For example, if you have a foot on a hold with your knee bent, stand up before reaching for the next hand hold. Maximize your upwards extension on the available holds before reaching for new hand holds. Keep your arms straight as you do this so as to not pump out your biceps. (There is one situation in which this is not good advice -- if your hand holds are slopers, it's often better to stay underneath or to one side of them.) This is strongly related to keeping your hips close to the wall -- if you do that, you will naturally keep your legs straighter.
Bumping is another technique to gain a few more inches. If the rock is such that the only hold within reach is too small to hang off but just a little farther up is a much more generous hold, you can bump. Move to the lower hold, and quickly use it to gain more momentum, so that you can reach the next solid hold.
Plan your route
As you are climbing, think at least one move ahead. Make sure that where you place your right hand now is where you want your right hand so that you can move your left. Same for your feet. It doesn't always make sense to grab the first available hold, especially if the climb veers to one side. If you are climbing to the right and moving your right hand, move two holds over so that you can move your left rightwards without having to match. This will make your climbing smoother.
Pay attention to the shape of the holds. If they are only positive (allow for a good grip) on the left hand side, then place your body to the right of the holds, so you can more effectively resist gravity.
Just enough force
Easy climbs tend to have holds with enough grip that launching oneself towards the wall can be compensated for by pulling on the holds. Not only will this stop working as the holds become smaller and balance more important, but it also wastes energy. Use only enough force to reach the hold, don't overshoot. (This is related to stepping lightly.) Similarly, use only enough force to grip the hold; don't waste energy squeezing the rock harder than is necessary to hold on.
Good climbers make climbing look effortless. One aspect of achieving this is maintaining momentum from move to move. If you are planning ahead, don't stop at each move; flow between moves.
A lot of moves in a climbing route may seem difficult at first, especially if you are lead climbing. Psychologically, this makes it difficult to fully commit to the move, since your body is tensing up in anticipation of falling. Remember, however, that all the equipment is there to keep you safe, and that you are usually protected from injury even if you don't make the move. Getting over this psychological fear - being bold on a problem - will help you focus on the climb instead of the fall.