Do you see the snow covered mountain to the left of Half Dome? That is Cloud's Rest. It is a VERY LONG day hike from Yosemite Valley. When I did the climb years ago, we started from Tenaya Lake on the Tioga Pass road. I would recommend that route.
Vernal Falls is a short walk, partly on a paved trail and then a quick stint on the wet Mist Trail, which is done every day by many visitors. If the Falls are full, take rain gear. If you are in good condition, you can continue up the same trail, past Vernal, to Nevada Fall. That is more of a back country experience but longer and more strenuous. Above Vernal Fall, there is a lovely vernal pool in the river that just begs out for the hiker to hop the fence and cool the feet. Please resist any impulse along those lines. Nearly every year we lose a few tourists who ignore the signs and get swept over the falls. There is no surviving that.
For further afield, I'd recommend the drive up to Tuolumne Meadows, on the Tioga Pass Road. A day hike through the meadows to one or more of the big domes is impressive and quite beautiful. We just had a big storm roll through and they had to close that road due to snow but it has re-opened as of last info. There may be some remnant snow on the higher points when you are there and weather has been unsettled so check conditions before heading off alone.
There are some very nice short day hikes on the Glacier Point Road. Taft Point is quite easily reachable and puts one on an overhang, overlooking Yosemite Valley. A bit further along the GP road is Washburn Point, which is right on the road and gives nice views up Little Yosemite Valley, behind Half Dome. At the end of the road is Glacier Point, which offers short walks around the point to various nice viewpoints.
For a nice, non-hiking day, you could drive down to Wawona and visit the Pioneer Center for a look at early life in the area. Another nice few hours, easily done from Wawona, would be the shuttle bus to the Mariposa Grove of big trees. If you have never seen the Giant Sequoias, you really should stop in for a visit.