Dust spots on the sensor are annoying, but not a tragedy. Cleaning off the sensor is serious business, so please exercise caution. Before you start the sensor cleaning process, carefully read and follow all of the required steps in your camera manual.
All SLR camera bodies feature a cleaning mode that locks the mirror up. If your camera can be cleaned while running on batteries, make sure that your battery is completely charged before you start the cleaning routine. Should your camera lose power midway through the cleaning process, there is a chance that the shutter mirror could come crashing down and get smashed.
Once you have the mirror safely locked up, the safest way to get dust off the sensor is with a hand-powered air blower. The Giottos Large Rocket Air Blower is my favorite.
Do not use an ordinary blower brush with bristles and never touch the tip of the blower to the sensor. There is no need to even put the blower tip inside the camera body. Just hold it close to the camera and blast the sensor with a couple of quick bursts of air. When you are done, turn the camera off, blow off the mirror, and then put the lens back on.
If air alone is not enough to clean out your camera I would suggest that you seek professional help. It is a whole lot less expensive to have a camera professionally cleaned than it is to replace a damaged sensor. If there isn’t a camera shop that you trust near you, and if you don’t want to send the camera back to the manufacturer for professional cleaning, then I suggest getting a Visible Dust Arctic Butterfly 724 Sensor Brush.
Here’s a video showing the Arctic Butterfly Sensor Brush in action:
Since camera cleaning is a bit of a chore, I suggest that you avoid changing lenses in dusty environments so you have to do it less often. I also suggest that you make it a habit to always turn the camera off before removing the lens or opening any of the doors. When the camera is turned off there is less electromagnetic charge around the sensor and, thus, less attraction for dust particles.
If you are careful about where you change lenses, and if you learn to clean out your camera body with an air blower, then dust buildup will be a minor annoyance rather than a major calamity.