The three most commons mistakes beginners tend to make as seen by coaches are the following:
1. Head position (any stroke). Your body has to adjust to however you position your head. If your head is too high, your legs go down causing more drag and resulting in "swimming uphill". The position of your head will determine the balance of your body in the water. Swimming unbalanced will not result in fast swimming. Streamlining is not just about positioning your arms above your head and squeezing your ears, it is about swimming streamlined, meaning you keep your head inline with your body.
2. Breathing. Exhaling is just as important as inhaling. A common mistake for beginners is that they will not completely exhale underwater prior to lifting/turning their head for a breath. This slows down the whole process and causes drastic changes in body position. To properly breathe, swimmers should exhale completely underwater (mostly through their nose) and inhale when turning their head slightly.
3. Short choppy strokes. Beginners often just try to move their arms as fast as possible. To get the greatest distance and speed it is necessary to get a full extension of each arm stroke. The deeper your hand is during the power phase of your stroke the better (the water deeper down is more still, think of pulling on a ladder as opposed to an escalator.
The following is a list of tips on how to improve in swimming.
If any of the tips don't make sense, don't worry or stress. Learning to swim is best accomplished by a combination of the technique being instructed and then tried and/or demonstrated.
Basics:
No matter what the stroke, work on the 4Hs. Try to get your Hands, Head, Hips, and Heels on the surface.
Have your swim buddy or coach demonstrate a stroke. Try to emulate (copy) what they are doing.
Focus on good technique and practice habits and you will be heavily rewarded. Work on your technique first and try to keep good technique throughout your practice/workout. It is better and more important to focus and establish good technique over just trying to go fast.
Like good table manners when eating, what you practice is what you will do in public (ie., meets).
Set personal, meaningful and purposeful goals. For example, when setting a personal goal time, try to set other smaller goals which will help increase your chances to achieve your larger goals. For example, the number of practices you plan to make, number of laps being able to complete during a practice, a particular stroke or flip turn technique, etc.
What you learn about yourself in swimming concerning how you commit, practice, relate, improve, compete and perform is growth and understanding which helps develop your entire self and can make you a stronger and happier person.
Don’t forget to include a proper diet, hydration (drinking water) and plenty of sleep/rest with your workouts and meets.
Proper etiquette is to not get out of the pool until everyone in your heat completes the race.
Beginners:
Learn to dive on a start, instead of jumping in. You can quickly improve your times by seconds.
Avoid touching or temporarily hanging on the lane ropes.
Adjust your googles well before you race to make sure they will not come off or cause problems during your race. It may be a good idea to practice with the googles you will use in meets.
Try to swim as straight down the swim lane as possible, since this will minimize the distance. For beginners this can be especially challenging on the backstroke. A way to stay in a straight "line" is to look at the solid black line on the bottom of the pool.
The first couple of weeks of practice can be especially difficult, so hang in there. Even for athletes in good shape from other sports, learning to breath properly for swimming can be a challenge.
Freestyle:
Keep your head down with your eyes staring towards the bottom of the pool. This naturally lowers your head and raises your hips and legs to reduce your resistance to the water.
Completely exhale while your head is in the water and exhale mostly through your nose. Breathe to the side with your head tilted slightly towards your armpit. Breathing to the front interrupts your stroke and forward movement. When breathing to the side practice keeping one google in the water. This will result in quicker breathing and help in keeping your head in a good positon to maintain balance in the water.
During the overhead stroke bend your arm at the elbow and enter your arms into the water at 11 and 1 o'clock, then make a long, slight S movement with your left arm and a long, slight backwards S movement with your right arm. The end of the S is very important and is the power phase of the stroke.
Swimming with long and deep strokes is sometimes referred to as "using your big arms".
Keep your fingers together, but relaxed.
Click here to see a QuickTime animation movie of Front Crawl.
Breast: 
Point your hands and fingers together and visualize a key-hole which your entire body will pull through.
Keep your knees fairly close together during your kick. With the breast stroke kick the hips should be behind you and not under you.
Pull on your breast stroke in what is sometimes referred to as "scooping the ice cream towards your face".
Keep your chin tucked while breathing.
The key thing in breast stroke is the rhythm - think butterfly, and swim breast stroke. Pull for power - kick for speed.
Click here to see a QuickTime animation movie of Breaststroke.
Back:
Keep your head back and face pointing straight up. This naturally lowers your head and raises your hips and legs to reduce your resistance to the water.
Like in freestyle reach towards 11 and 1 o'clock. Reaching too far such as 12 o'clock will create more difficulty in getting to the power phase of backstroke. Rotate your palms pointing to the outside. Remember: “thumb, ear, pinky”!
Learn to use the overhead flags to determine how close you are to reaching the end of the pool for turns and finishes.
The one stroke where you are able to breathe more freely during the entire stroke. Don’t rob yourself of air.
Click here to see a QuickTime animation movie of Backstroke.
Butterfly: 
The legstroke move consists of an 'upbeat' and a 'downbeat', that are started out of the hips.
The upbeat takes twice as much time than the downbeat.
Inhale using your mouth and exhale out using your mouth and nose.
On the dolphin kick keep your feet together for added power (and to also avoid a DQ).
The pull over is above the waterline, through which the resistance is reduced to a minimum.
It is recommended to breathe after every 2 armstrokes. As a result of this way of breathing, the frontal resistance is reduced to a minimum.
Frequently made mistakes on the Butterfly
Click here to see a QuickTime animation movie of Butterfly stroke.
Racing:
Stay focused on your own lane and performance. Looking over a lane towards another swimmer, even slightly, during a race will slow you down.
Learn to touch cleanly and quickly the finish wall. For races with touchpads, on the finish be careful to touch the front of the pad and not the top. On finishes, imagine your swimming through the wall not to the wall.
Try to hold your breath inside the flags and not take the final breath before touching the finish wall.
On an open turn, after touching the wall, have your hand move as close by to your ear as possible. A flopping arm, over and out from your head, is not as quick as one moving crisply and closely past your ear.
Advanced:
When you breathe on freestyle, concentrate first on breathing to the side, then on having your mouth parallel to the water, instead of over the water.
Several different types of breast strokes have evolved and are now used competitively. Discuss with your coaches the different variations and what type may be better or best suited for you.
Open Turn: Quickly touch the wall with one hand above the other on a turn. This helps you change your momentum more quickly on the turn. You can try a dry-land drill of practicing quick touches against a wall. Remember to have your hand move closely past your ear and to not loosely flop over.
Flip Turn: On turns, there is an optimum length to make the turn between not swimming too close (swimming farther) and close enough to allow a good push off with your legs. Have your legs separated and shoulder width apart when they are on the wall to maximize your power and strength on the push off.
On turns, avoid taking an extra breath as you go into and away from the wall. This takes a bit more practice and stamina, but you can shave quite a bit of time off your race. It is difficult, but try getting at least two strokes when coming away from the wall too.
On a flip turn, exhale out of your nose during the turn to help prevent water from going up your nose.
On a backstroke start or flip turn, a trick some swimmers use is to curl their upper lip to cover their nose. This helps prevent water from going up the swimmer's nose and allows them to stay under the water longer (versus exhaling out the entire time).
On dives, there is an optimum depth to take for extending out and coming up on a start.
Practice your relay starts, knowing that you can begin your momentum and lean out in your dive as your team mate touches the wall. Your toes just cannot leave the platform until your team mate touches.
To swim fast, no matter what stroke, it comes down to three key things. Reduce the resistance, reduce bad drag and move through the water as efficiently as possible. Remember to maintain balance!
Note: You will need the Quicktime plug-in to view the movies. You can get it for free from Apple Computer for just about any computer platform