From my own personal experience of having spent a week just in the park a few years ago, I'd offer the following "rough" ranking of places to see:
1) Grand Canyon of the Yellowstone: a) Red Rock Trail for a close, awesome view of the Lower Falls; b) Uncle Tom's Trail for the breathtaking vertical ladder down for even closer view of the Lower Falls. If you have any heart conditions, then avoid it; and c) Artist's Point, the most photographed spot in the Yellowstone, period. I thought I was in a dream when I was at this particular spot.
2) The Old Faithful (for the incredible Inn itself and of course for the 10 minute interval eruptions)
3) Hayden Valley to see wild animals -- bears, pronghorn deers, bison, etc.
4) Mammoth Hot Springs: Lower and Upper Terraces. If you choose to visit this area, then stop by the "Boiling River" nearby for thermal soaking.
I could go on and on with the list, but the above are the ones I'd recommend given your short duration of stay, particularly the number one.