You should expect to still see some snow on the sides of the roads at that time, but the weather should be otherwise clear. Almost everything in the Park should be open.
It will still be a bit early for peak bug season, though the ground on some hikes can be soggy.
Several trails will probably not be open yet, either.
You will beat most of the traffic. I went last May, and despite so little of the Park being open, it was so fantastic to drive without having so much traffic.
The weather will likely be cool but not intolerable. Expect very cold nights, but since you are staying indoors, it shouldn't be a problem.
One thing you should be wary of in Yellowstone is the dramatic changes in weather. Always have rain gear nearby, even on the sunniest of days. The weather can change in a flash. I once got stuck on the boardwalks at Old Faithful on a sunny day with only a few puffy clouds on the horizon. In 30 minutes (I am not exaggerating - go here at http://www.yellowstone-online.com/images/heartcastle.html - this was taken before we got hit), we were being drenched in a cold, driving rain nowhere near shelter. Be careful of that, especially with a family...at least have an umbrella with your family at all times or a couple of plastic ponchos.