You say you want to see Boston, New York, and Washington DC. That could take up most of your month right there. If you like museums and history, allow at least a week in Washington! And you might want to squeeze Philadelphia or Baltimore in there too. You don't say how you will be traveling. One of the things I envy about the northeastern part of the U.S. is that they actually have decent train service, so you could travel between all these cities by train. That is generally not practical in the rest of the U.S. You will either have to fly between cities or spend days in a car, so I would choose 2 or 3 additional areas you want to visit and spend a week in each one. (Thirty years ago I traveled by bus, but I tried that again about five years ago and it was "the trip from hell"; I will NEVER recommend bus travel to anyone!). Once you get to your destination, public transportation is generally poor, so plan to rent a car in each city that you fly to. (Washington DC is an exception; take the Metro there.)
If you go to Georgia you may want to go another 400 miles south and visit Disney World in Florida. If you do, plan to also visit Daytona Beach and the Kennedy Space Center; you might also want to travel to the Gulf side of Florida (maybe around Clearwater?). Keep in mind that Florida is extremely hot in August, so you may have to plan your sightseeing for the morning and evening, with a stop somewhere air-conditioned in the middle of the day. On the positive side, there are sometimes very good prices available in Florida during the last part of August; I do not know if it is because it is so hot or if it is because all the children in the south have already gone back to school by then.
There's also a lot to see in the middle of the country, but because your time is short, my next recommendation would be to fly to Salt Lake City and drive to Yellowstone. You'll see some spectacular geological formations and scenery, and you can do some of the "old west" stuff like ride horses and go on a chuckwagon cookout. Then, if you really want to see LA, you can spend your remaining time in California. I have never been to Los Angeles but I have spent one day in San Francisco and it was beautiful. I only had time to drive over the Golden Gate bridge and up into some of the farm country, then that evening I rode a cable car and visited Chinatown, but I know there is a lot more than that to see. I also believe that Las Vegas, with its gambling casinos and shows, is within driving distance of Los Angeles, but someone from the west coast will have to confirm that.
Since you are a L&C fan I hate NOT to recommend visiting Kansas, but as someone who once routinely drove 400 miles across that state each week, I do not recommend it unless you are VERY interested in agriculture. (I was, but most people are not.) Do take a look at it when you fly over it; the pattern of fields, laid out in square "sections" of one mile on each side, makes the view interesting.
Keep in mind that most of the cities you have mentioned can be quite expensive. I hope you will find friends to travel with, because if you cannot find someone to split hotel costs with, you may not have enough money for sightseeing and/or transportation. On my recent trips, even to fairly low-cost areas, I have had trouble finding a hotel room for less than $80 to $125 per night, and I know cities such as New York are much more expensive than that. It is best if you can book your hotel rooms in advance on the Internet, because you will get better rates than someone who walks up and asks for a room. There are some alternative solutions such as college dormitories and youth hostels, but I have not used any of those lately so I cannot offer any advice there.
I could go on forever, but I will stop now and hope that what I have written so far is of some use. I hope it works out well and that you have a great trip.
