A few brief halakhot about snow (some of this is taken from an email I sent last year):
Snow, even if it falls on Shabbat, is not muktzeh or nolad (see Shulkhan Arukh OH 338:8 and Mishne Brurah, #30; see also Shmirat Shabbat ). However, one cannot make or throw snowballs or make snowmen or snow angels (ibid).
As to sledding - there is not a technical Shabbat prohibition against this. The grooves creates by the sled are no different than the impressions left by one's shoes when walking. However, such activity, especially when it involves a lot of exertion and movement, could likely be in the category of uvda di'chol, weekday activity. It is like playing ball on Shabbat - is it a nice relaxing Shabbat activity, or is it a weekday, very un-Shabbos activity? Context matters. Certainly, the greater the exertion, the greater the problem. And if it immerses one into an experience where one forgets or loses the sense that it is Shabbat, then it certainly is uvda di'chol and should not be done. On the other hand, transporting small children to shul via a sled, especially if there are no reasonable alternatives, should be fine. Because context is critical, please consult a halakhic authority.
More importantly, regarding safety, one can put salt or sand on snow or ice on Shabbat (OH 320:14, and MB 41, and MB 318, note 107) and, if there is any concern that someone might slip and hurt him or herself, you should make sure to do this. In such one can even use sand that has not been pre-designated for this purpose and would be muktzeh (OH 308:6).
As to shoveling snow, Rav Tzvi Pesach Frank (Har Tzvi, Tal Oros, Soter) rules that if the snow has hardened, shoveling could be considered a form of boneh (making a path) or soter (of the present snow). The only other issue is that we don't do activities that require exertion (tircha) on Shabbat, but this would be allowed for the sake of a mitzvah (getting to shul) or if there is any danger or risk by leaving the walk unshoveled. All of this refers to shoveling a sidewalk or driveway. Shoveling off of dirt or grass raises other concerns (smoothing out the ground).
Above all, be safe. It is far more important to stay home and be safe than to go to shul and slip on the ice and risk serious injury.