Hmmmm, 3-4 days is a short time to stay in Yellowstone. Yellowstone is my favorite location on the face of the earth (and I've seen a lot of the world)!!!!!!! When are you going to Yellowstone? Do you want to stay within the park itself? Lodge, cabins, camping, hotel? The time of year you choose to visit is important in determining where you can stay. Many of the lodges, cabins, etc close up after Labor Day and don't open again until after Memorial Day.
Yellowstone is enormous and most people don't realize how many miles you need to drive to get from point A to point B if they have never visited before. The figure 8 ring road is 184 miles and driven at an average of 30 miles an hour. In in summer, with heavy traffic, the drive is slower. There is always road construction going on within the park because the snow covers the roads at least 7 months a year.
Asking what the vital areas to see in Yellowstone is like asking people their favorite breed of dog. . . everyone will tell you something different. Here is a list of my own personal favorites of Yellowstone, moving from north to south and then again movng from west east.
Mammoth Hot Springs, Lower (Norris) Geyser Basin and Upper Geyser Basin (area where Old Faithful is located) and West Thumb, Grand Canyon of the Yellowstone which include the lower and upper falls. This is what I would concentrate on seeing in a 3-4 day visit. Also, I would choose to go during elk breeding season in mid September (just after the summer season has ended) and stay in the cabins at Mammoth Hot Springs. I absolutely LOVE being able to open my cabin door in the mornings, step outside and see elk wandering through and the steam from the geothermal activity going on in the park. AND, as my husband told me that I would be, I LOVE the smell of Yellowstone with all that sulfer in the air. Then, if at all possible I would plan on a return trip for during the winter. My husband and I got to go during February one year and there is NOTHING in the world like standing watching Old Faithful erupt without a single living soul in sight! Dead of winter is the only time you can experience this.

Before going check to see which entrances are open. North entrance may be closed. We were there this time last year. North entrance and another route were closed. We stayed in West Yellowstone which is at the west entrance to the park. Grizzly bear and wolf discovery is there. We had the best time there. We stayed across the street from the discovery, our windows faced the wolves and we could hear and see them. What a great place. Most of the shops are closing there for a short period until the snow moves in. You can stay in the park at the hotel there but personally, I've been there 5-6 times and we either stay at the west entrance or north entrance. Make sure to use facilities whenever and wherever you can. It can be a long drive to the next open facilities. This is an enormous park. Make sure you stay in the car, do not get out. Buffalo can run fast. What route will you be taking in? If you'll approach from the west, stop in Cody, Wyoming. A great little town. If your coming from the southern route, make sure to see the Grand Tetons, they are magnificent. You will no doubt see snow on the mountains.
Hope this helps. Drive carefully. Use your lower gears when going down, it makes a big difference and you won't burn up your brakes. Have fun.

thanks so much for your help. We thought we would go in May but I have read that some places arent' even open - we wanted to go before school lets out (before all the tourist). Anyway, I thought it would be fun to stay at yellowstone. WE are coming from Virginia so it's a long trip and we may stay longer. Yes, I want to see the Tetons also. This country is so beautiful but we can only go so many places. From Yellowstone, we are going to San francisco... we are retiring and this is our first trip across country - never have been to Wyoming or california or anyplace west so it will be a real treat. We went to Wisconsin one year and stayed at a horrible place and didn't want to make that mistake again. I bought two books today so that I willbe somewhat familiar. Thanks...let me know if you think of anything>

May is iffy, due to weather may still have snow. My SIL stayed at West Yellowstone and took 2 tours and really enjoyed it and was able to see quite a bit in 3 days. Lots of motels in West Yellowstone and a mc donalds too. Great bakery also. You could take a tour to the Tetons also. Not far from Yellowstone but easier to get to by car. Go to Bozeman Mt, then go down Hwy 85, turns into 191 and watch for sign. Do not take the road from Livingston, very windy and my be closed. If you belong to auto club or know someone who does, you can get maps and information. Even if you don't belong, joining for one year might be worth it for the information for such a long trip. Free maps and advice. Again Tetons may have snow/ice etc. Check for tours in area. Don't forget Yellowstone has had 2 major fires so some areas my be off limit. Need more information let me know
Marie

When we go from Calif (near Los Angeles), we go north of Minot ND in 3-4 days pulling a trailer, so we know all sorts of short cuts.

Be sure to see Jackson Hole, WY. It is a fun but pricey place.