Most casual/amateur photographers believe that capturing dramatic action photos can only be accomplished by professional photographers armed with Pro-level DSLRs and fast (and very expensive) telephoto zooms. The truth is that professional photographers have to get the shot, or they don't get paid.
Using the very best equipment they can afford is insurance to radically improve their probability of capturing that "killer" shot. Almost anyone can shoot excellent action photos though, and even simple P&S digital cameras are capable of doing a good job. A friend of mine had only shot pictures with his cell phone up until he bought his first digital camera (a Canon SX50 HS) a year ago. He accompanied me to the local skate park a couple of times for shooting tips and to practice action photography. By October of this year, he was using that Canon SX50 HS to shoot local high school football games as a stringer for a small town weekly newspaper.
There are only a few main factors that separate good photographers from great photographers and they are - experience, the ability to dramatically frame the subject of the photo, and timing--the ability to capture the "decisive moment". And amateur photographers using P&S digicams can master all three, with practice. It doesn't make any difference whether your primary interest is sports, skateboarders, rodeo riders, dancers, swimmers, or ice skaters--the pathway to success is the same--practice, practice, practice.
The bottom line in action photography is to be in the right place at the right time, so select your shooting locations carefully. In youth sports like soccer and little league baseball, adult amateur sporting events, skate parks, outdoor ice skating, and local sledding venues (in new snow) shooters can easily position themselves close to the action. With high school and college sports, formal dance recitals, gymnastics, and professional sporting events, it is much tougher to get close to the action. If you want to shoot dramatic action photographs where access is limited, try asking local coaches or dance directors or gymnastics instructors if you can take photographs during practices. Get as close as you can without risking injury to yourself, to others, or potentially interfering with the action. Photographers looking to capture dramatic action shots in my hometown are lucky because we have a professional quality skate park. Lucky for me, photographing skateboarders and BMX bikers is one of my passions. Research the action photography venues in your hometown and I'm sure you'll find numerous locales for shooting action pictures.
The first step is to thoroughly learn your camera's capabilities. Most current P&S digital cameras include a sports/action scene mode. Enable this function and your camera will automatically customize all exposure parameters to help you capture dramatic action photos. If your digicam doesn't provide a sports/action scene mode, just put the camera in program mode. Watch for a while and learn to follow the action. Shoot a lot of pictures as the action unfolds. A dramatic action photograph should tell a compelling story, so you will need to focus on capturing the quarterback catching (or throwing) a pass with the ball in the frame, close-in shots of swimmers resolutely slicing through the water, or skateboarders in mid-air, or a ballerina in the middle of a toe tip pirouette. Finally, don't become too obsessed with action--always keep an eye open for captivating environmental portrait opportunities like the coach signaling his players from the sidelines, or the tired skateboarder using his skateboard as a pillow while he rests up for his next run, or the disappointed BMXer tightening the chain on his bike for that tiny bit of extra control, or the hyper kid sitting on his sled while he waits for his turn on the snowy hillside.
Timing is the most important factor in action photography and it is also the hardest to master. Capturing the moment of peak activity is strictly a matter of practice. The more you shoot, the better you will get, so don't be discouraged when you repeatedly miss the decisive moment. Instead, try to figure out why you missed the peak action moment and refine your technique. If you can learn to anticipate the decisive moment, you'll succeed beyond your wildest dreams. What I mean by this is if you are shooting a skateboarder and you know the kid is talented and a bit of a show-off --wait until he begins his most dramatic trick to push the shutter button. If you are shooting the quarterback, wait until his arm reaches its maximum rearward extension and push the shutter button at that instant and you'll capture the ball leaving his hand. With a little practice you'll discover that you can nail the peak action moment almost every time.
The second most important element in shooting great action pictures is framing and this skill is relatively simple to master. Your subject should fill the frame with only a bit of their external environment around them. When you want to highlight a single subject, he/she should be facing the camera and you should be very careful to include a bit of headroom in the shot. Action photographs with a single subject should generally be shot in vertical format. Use the horizontal format for showing action in team sports or group settings or if you want to isolate a subject for impact. My final piece of advice (other than the admonition to practice) is to edit your images ruthlessly. If your picture is blurry, cuts off part of the action, misses the peak action moment, has the subject facing away from the camera, or leaves too much space around the subject--delete it. Save only your very best pictures. Good luck and good shooting.