Assuming they don't reopen Tioga Pass next week, you can only enter from the west, either highway 120, 140 or 41. So depending on where you are coming from, I would take the most direct route. The only variation on this is if a storm is forecasted, it might be easier to 140 as it is at a lower elevation and much less affected by snow.
With a dog you are rather limited for hikes, pretty much just paved trails and roads on the valley floor. For more information see:
http://www.nps.gov/yose/planyourvisit/pets.htmYou could walk up to Mirror Lake (1 mi, mostly flat with a 200 ft gain at the end). The lake will be mostly dry at this time of year, but you are right below the face of half dome and have great views of Tenaya Canyon.
Otherwise do sections, or the entire, valley bike loop. I think the full loop is ~ 7 miles, but you can break it up into much shorter sections. For great views of the valley and yosemite falls, finally running again!, do the section(s) by swinging bridge, the chapel and cooks meadow. You can even combine that with the walk to lower yosemite falls. For this one you can start right at the visitor center and altogether is about 2 miles.
No snow yet in the valley.