I'm not sure you understand the magnitude of the trip you have planned. I think it is far too ambitious, especially for someone with limited experience.

First of all, you seem to have no idea how much driving is involved, how much time it will take or how much it will cost. I've driven from Ohio to Texas a couple of times (takes two very long days with no stops except for gas and 1 overnight) and from Ohio to California and back, plus a lot of other cross country trips. There are websites now that will give you the distance, time and estimated costs. Use them to see what you are up against. It is not easy to drive 500 miles a day. I've done it many times. You need a reliable and comfortable car. Do not rely on GPS for shortcuts. Too often the roads it will put you on are not somewhere you want to be. Join the AAA or another auto club that provides maps and breakdown services.

Second, if you plan on camping or staying at a hotel or motel in a park, most, if not all of these parks require reservations in the summer. Yosemite is definitely one. Waiting until you get there will likely mean staying outside the park. Start making reservations now.

Third, the forums on this site are not really the best way to learn the basics of camping. Start reading books like Backpacking for Dummies or The Complete Walker. Your local library should have them or something similar. At this point, you don't even know what you don't know. I could give you a gear list, but it would be useless and misleading. There are general gear lists on this site and dozens of others.

May-June sounds like summer in most places and is, but in the mountains, you might still encounter snow in some places. Spend some time on the websites for each of the parks you want to visit, get the weather records which they should have posted and plan accordingly. Here are examples-
http://www.nps.gov/deva/planyourvisit/weather.htm
http://www.nps.gov/yell/planyourvisit/weather.htm
http://www.nps.gov/yose/planyourvisit/tiogaopen.htm

What gear and clothes you need are weather and activity dependent. As long as you no further than about an hour from your car or shelter, you can get away with cheap gear and a minimum of clothes, any further than that, you need to be prepared with clothes and equipment sufficient to enable you to survive overnight.

BTW, if you think I'm being a wuss about your plans, my experience includes travel in South America, New Zealand, Australia and the Western US.