I'm a planner by nature and almost always get reservations, That said, you probably won't need them if 1) you are not planning to camp in a state park on a weekend and 2) if you will be getting to the CG before 5:30 PM. (I'm not addressing Yellowstone in this paragraph, as you already have reservations there - a good thing.)
As for Yellowstone, I believe you have another thread about routes to use, and you have gotten advice to go in via east entrance, as you're coming from the east. Just FYI, from the east entrance to Madison it should take you about 1 1/2 to 2 hours, if you don't stop for more than one or two short photo ops. There may be some construction on the road between east entrance and Fishing Bridge, just because there usually is some work needed after winter in that section of the park (one of the highest & snowiest) - if so, that will add a bit to your time.
Enjoy! It's a great time of year to visit. And unless we get some big late-season snows, the roads should be clear through most of the park. A few sections are scheduled to open - weather permitting - during the time you'll be there.