We just got back from a spur of the minute trip to Yellowstone last weekend. We talked about it on a Wednesday night and left that Friday from Southern Califorinia. We went up thru Salt Lake, into Idaho and entered Yellowstone from the west entrance. On the return trip we came out thru the south entrance thru Grand Tetons/Jackson Hole.
A couple of things to note - Highway 191 between Grand Tetons and South Yellowstone had a section (maybe 10 miles) that was being re-paved. Nothing major, but you do need to slow down and a couple of sections were a bit bumpy. The road was mostly compacted dirt and there were at least two sections where we needed to wait for an escort. 191 was a nice drive with a lot of places to stop and take a short walk/hike (Lewis Falls, etc).
In Yellowstone there is a figure 8 loop called the 'Grand Tour'. The day we arrived in the park, the road between Madison Junction and Norris Junction was closed and will be thru Spring. If you are coming from/going to West Yellowstone, this eliminates the shortcut thru the middle of the park and can make seeing the major attractions take quite a bit longer....though we enjoyed seeing more of the park taking the 'long way'. If you're at fishing bridge it won't impact you as much because you will be in the center of the park.
Road closure info can be found here:
http://www.nps.gov/yell/travelalert.htm
Because it was a spur of the moment trip, we stayed at the West Yellowstone KOA in Montana (about 6 miles from the west entrance). We were very happy with this park as base camp for 3 days - though the price was a bit steep at $71/night.
I think our favorite area of Yellowstone was the Canyon area and the drive up to Mammoth. Saw lots of wildlife and the views are spectacular (the fires in 1988 have really changed the park and areas where all you could see was thick forrest now give way to younger growth and spectacular vistas).
At the Grand Tetons we took the boat shuttle across Jenny Lake and took a hike up to Inspiration Point - whole family enjoyed it and in Montana we took a horse-back ride that was great.
We saw the tour buses, but preferred to take things at out own pace, especially with two boys (8 and 5) and a dog.
Don't think you can go wrong - have a great trip!