Hacham Ovadia Yosef, in his work Halichot Olam (vol. 4, p. 265), discusses the Halachot concerning watches on Shabbat. He writes that if a person has a watch that runs without batteries, it is permissible to adjust the time of the watch on Shabbat, and also to wind the watch so that it will continue running and not stop. However, if the watch had stopped, then it is forbidden on Shabbat to wind it to restart its operation. Winding a watch that had stopped would constitute "Metaken Mana" (fixing a broken utensil), which is forbidden on Shabbat. As mentioned, however, if the watch is still ticking, then one may wind the dial to prolong its runtime.
By the same token, it is permissible on Shabbat to wear a watch that is operated by the movement of one’s hands. Since the watch is already running, one may wear and walk around with the watch even though every movement he makes with his hand has the effect of extending the watch’s runtime.
It is forbidden, however, to adjust a battery operated watch on Shabbat, since this entails discontinuing and then restarting the watch’s mechanism, which runs on electricity. Battery-operated watches may be worn on Shabbat, and we do not consider the watch a "Bassis" ("base") to the battery such that it should be considered Mukse. Even if the watch has buttons that perform certain electric functions, such as a button to illuminate the screen, it is permissible to wear the watch on Shabbat. Hacham Ovadia writes that we cannot legislate a new prohibition against wearing such watches out of concern that one may press the buttons. Thus, it is permissible to wear a battery-operated watch, though it is forbidden to press the buttons or to adjust the time.
Summary: It is permissible to adjust the time on a watch that does not run on batteries. One may wind such a watch to prolong its runtime, but not if it had stopped running. A battery-operated watch may be worn on Shabbat, but it is forbidden to press any of the buttons or to adjust the time.