The entrances to the park do require you to drive through a mountain pass with sharp bends. During the snow season, tire chains are required. (http://www.nps.gov/yose/planyourvisit/chains.htm) If you aren't comfortable driving in those conditions, I'd recommend that you take the shuttle. The Park Service provides a park from towns just outside of the park entrance and within the Yosemite Valley. Here is their page with more information: http://www.nps.gov/yose/planyourvisit/bus.htm
Driving to the park border will be easier. The roads are bigger and more likely to have maintenance if there is snow. It's been a while since I've driven there, but as I recall, the roads leading to the park are not very mountainous or narrow.
Be aware that during the winter months, some of the entrances to the park are closed. So be sure that you don't go to one of those. I beleive Mariposa is the most popular entrance and is open year round unless there are extreme weather conditions. Here is information on the typical closures during the winter season: http://www.nps.gov/yose/planyourvisit/winter.htm
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