First of it would help if we knew what you were shooting pictures of.
If your subject is static then you wont have to concern yourself with motion blur from the subject. But you will need to be concerned about it from the camera. there is an old saying "Your first lens should be a tripod" Meaning that a tripod is the most important thing you can own to achieve sharp images. It will allow you to use slower shutter speeds and stop the lens down for greater depth of field.
Unfortunately many modern AF zoom lenses do not have a distance scale or a depth of field scale on them. But if your shooting landscapes you might consider getting a wide angle lens that does. This way you can use the hyper focal distance for maximum depth of field. I will explain this below if you care to know.
For other subjects you are stuck with using the smallest aperture you can get away with and hoping for the best. But again a tripod can help. A small amount of subject movement is not as bad as camera shake.
NOTE: this must be done in manual focus
It's simple, the focusing ring on the lens will have a scale of feet and meters. On top of the lens will be an indicator to tell you the focus distance. To the left and right of that indicator will be F/numbers. That is the depth of field scale. It will look something like this.
16 11 8 5.6 * 5.6 8 11 16
The number on the distance scale that is next to the indicator is the focus point . Every distance on the focus scale that is between the numbers on the DOF scale representing whatever F/number you are using will appear do be in focus. Now lets say you are using F/ 16. Focus on the farthest thing you want in focus. ( normally for landscapes this would be infinity ) now move that distance from the indicator to the 16 that represents the far end of the scale. Now you have the maximum focus and depth.