Here are some cities you may want to visit ---
Start in Boston, a beautiful American city. A lot of American history is here plus loads of great restuarants. It is a very walkable city - no car needed. Stay in the city not the outskirts. The hotel costs will be higher, but will be offset by not needing a car. Plus, you really don't want to drive in Boston. As an American, I don't like driving in Boston.
From Boston, you can take the train to New York City. Loads to do here, from museums, parks, the Statue of Liberty, Broadway and so on. Again, stay in the city and take the subway and/or taxis. Plus, you can take the double decker sightseeing buses on a multiday pass to get you around. You can still find some good deals on hotels if you look really hard.
From New York City, you can take the train to Washington DC. Stay near the "mall", which is the area around the Smithsonian museums (all free), where you will spend a lot of time. From here, you are in walking distance to almost everywhere you will want to visit. Again, the hotel costs will be a bit higher to stay in this area, but this will be offset by travel expenses and additional time needed to get and forth. If you can, take a night tour of the various memorials. Also, the national zoo is free and easily accessible via the metro.
You will need to fly to get to Louisiana. I'm not sure why you want to go there, as New Orleans is still recovering from the hurricane and it is definitely not a family destination. To be honest, it's not my favorite state.
Going out west, you will have to fly to Phoenix to get to the Grand Canyon. My suggestion is that you may want to take a tour from Phoenix rather than making the drive yourself unless you are comfortable on US roads. If you do drive, leave the car at Williams, AZ and take the train from there. It is a beautiful train ride to canyon from there. I would recommend staying at the canyon at least a night or two. However, you have to make reservations very much in advance. There are many high end resorts in the Phoenix/Scottsdale area for the pre and post canyon visit. However, their prices fall signfiicantly during the summer, so you might be able to find some excellent deals if you are there at that time. Be forewarned, it gets really hot in Phoenix during the summer.
Out west, San Francisco is a beautiful city with many things to see and do. However (I mention this because you are bringing your children and I do not know their ages), I'm sure you are aware that SF has a very large and open gay population. It is close to impossible to get tickets to Alcatraz unless you go through a tour operator. Again, stay in the city - preferably near the wharf. You really do want to avoid driving in the city -- plus hotels will charge you $20-30 per day for parking.
From SF, you should make the drive down to Los Angeles via the Pacific Coast Highway. This will take some time, but it is one of the most beautiful drives in the states. Carmel, Monterrey and Santa Barbara are really beautiful areas.
In Los Angeles, you have the film/television industry - where many studios will provide tours. You will need a car in LA, as the city is very spread out and public transportation is not really available/usable. You can also visit Hollywood, Beverly Hills, Bel Air, etc. There are other things to do there as well, but to be honest, I'm not much of an LA fan.
I hope this is helpful to you and hope you have a great trip.