Agree with the above posts, especially Cody and the museum and would add a couple more points. First, it can snow any month of the year, and has snowed on us once in June. Enough snow to cause the Park Rangers to shut down the part of the park we were in (Canyon), and for my kids to build a snowman. We were "stranded" only for a day, but it can happen without much warning. Secondly, it can get cold overnight even in summer. Last summer we experienced nightly freeze warnings in the park and West Yellowstone, again in June. So, take some winter gear with you. When you go in May, expect that not all facilities will be open, including some dining areas.
Best wishes for a great trip.
Bob