I agree with Daawgon, sticking with one or more regions close together would be a better bet. I think with only 18 days, since you are on the East Coast already, you should stay on the East Coast. There are lots of things to see on this side of the country. You can get a 15 day pass on Amtrak, valid for unlimited travel for a 15 day period for about $389 USD. Starting in Miami, I would suggest traveling to Orlando, Florida to see Disney World. After that, head up to Savannah, Georgia. (This is my hometown, so if you want to know what to do there, message me and I can give you a lot of good tips) From Savannah, take the train to Charleston, South Carolina (gorgeous city), then on to Washington, DC (this is where I currently live, so again, I can give you some good tips of things to do). After DC, you can head up through Philadelphia, Pennsylvania to New York City (stay as long as you like, there's loads to do in New York). After that, head up to Boston, Massachusetts then to Portland, Maine. (The coast is absolutely breathtaking, from what I've seen. I'll be going to see it for the first time in person next month). That should keep you busy for those 15 days, and the longest trip is about 9 hours between Charleston and Washington DC. If you see something you like in between, break it up. You will never be on the train for a very long period of time.
If you decide on Savannah or Washington, DC, seriously, let me know. I'll tell you all the best things to check out. I hope your trip goes well. I'm sure you'll have a blast, and welcome to America!
[ Edit: Edited on May 31, 2008, at 7:10 AM by lwboykin ]