TtT. Unfortunately dust is just a fact of life with DSLRs. Many nowadays have sensor cleaning which discharges them and "shakes" the sensor clean. Even these, though, are prone to the odd speck. For old cameras you can get a liquid cleaner that is applied with a sterile swab (don't use blowers!). It can all be rectified in Photoshop or a similar package. Don't lose heart.
Regarding the shutter speed I'd never go below 250 for general shooting. That gives a moderate to good amount of clarity. In the end it is all about practice and learning technique. The great thing about digital is that you can do that over and over wthout cost!!! Good luck. RR