Unless you want to make a fast trip down to Florida save the southern states for another year. It gets very hot and humid early in the year and by summertime it's pretty oppressive. The weather in April,May and June should be good in New England, the eastern seaboard and any states above the Mason Dixon line. July and August are best spent in the mountain states out west or in the Pacific Northwest.
I read a blog written by a couple from Israel who spend every summer in the US. I believe they're members of the forum. They have a fifth wheel which they leave in a different place every fall so it's ready to go to their summer destination without a lot of cross country driving. You may find their site helpful for choosing RV parks and activities.
http://danyshula.blogspot.com/ I second going to as many national parks as possible. The pass is great and will give you free admission to many federal sites besides the national parks.
Avoid making reservations that will keep you on a strict time table. There are so many things to see and do that it's hard to plan out each step of your journey ahead of time. Most RV parks do not fill completely and it's even possible to get a site at some of the most popular national and state parks because of cancellations.
Here are a few of my favorite planning sites:
http://www.uscampgrounds.info/ https://www.campendium.com/ http://www.fhwa.dot.gov/byways/ Have fun!