http://www.nps.gov/yose/index.htm) to find out which roads are closed and which roads will require snow chains. Snow chains, when needed, are enforced, so they simply won't let you through if you don't have them. They're not too expensive and can be bought in Walmart or other smaller supermarkets.
If you only have 1 day, you really shouldn't plan to do very much. Tioga Pass, Badger Pass, Mariposa Grove are all out of the question. Just head straight to the main valley floor, and visit the sights there, like Yosemite Falls and Angel Falls. Walk around Cook's Meadow and take in the views of El Capitan and Half Dome. Drive to the scenic spots which are marked on the official map as "valley view" and "tunnel view", they give you the standard postcard views of Yosemite Valley. If you maximise your day by getting there very early, maybe you can squeeze in just one short hike (and you'd better walk fast, no time wasting), I'd recommend mirror lake. Do take note that it gets dark early in a valley.
And you can check the ranger program schedule to see if you could join any of the guided walks if you'd like to learn more about Yosemite. Normally I'd strongly recommend them, but if you only have 1 day to rush around, it may not make sense.