Make sure you bring your passport. U.S. citizens can stay up to 90 days in the Czech Republic without a visa.
Pick a train. You will will probably be leaving from Budapest and arriving in Prague. There are at least 6 trains a day that make that trip: the Metropol, the EuroCity Commenius, the Hungaria, the Csardas, the Amicus and the Pannonia. Some trains make more stops, so duration is variable.
Take Hungarian currency with you when you purchase the ticket. The trip to Prague from Budapest will cost somewhere between 5,000 and 10,000 forints. Only forints will be accepted.
Decide when you'd like to depart. The trip to the Czech Republic will take around 9 hours; approximately 6 trains a day connect Budapest to Prague. Refer to the official timetable Web site for the schedule (see Elvira in Resources).
Find out if your train of choice offers different classes of travel. First-class travel is often recommended to travelers, but second-class travel is the common choice of Hungarian and Czech locals. The difference between price is slight, but the choice is yours.
Go to a ticketing agent at one of Budapest's 3 train stations or the state railway's central ticket office. The ticket office should be more accommodating to travelers who don't speak Hungarian.
Be very clear and thorough when booking the ticket. A small miscommunication then can cause huge problems later. You may want to ask which station your train will depart from, since Budapest has 3 train stations.