One of the things that makes Ayutthaya a popular day trip from Bangkok is the relative ease with which you can get there. The basic options are bus or train, and there are also packaged day trips available.
Buses leave Bangkok's Mo Chit bus station about every 20 to 30 minutes. There's no need to book in advance. Just show up and buy a ticket for the next bus. You can take a taxi all the way to the bus station, or take the subway to Kampaengphet station, and then grab a tuk-tuk or taxi from there to Mo Chit, which is very close by. The fare for an air conditioned bus is around 50 Baht (1.34 USD) and the trip usually takes under an hour.
Trains depart Bangkok's Hualumpong station more or less hourly. Third class (non air conditioned) fare is 20 Baht (0.53 USD). In the cool season from November to February, third class can be quite comfortable, with a nice breeze blowing through the cars. Trains with air conditioned second class are less frequent and cost significantly more. Trains take a bit longer than buses, from an hour to an hour and a half, depending on the number of stops, but I think the train is a more pleasant way to go. Ayutthaya's train station is just across the river from the old city. From the station, you can walk down to the river and catch a ferry across the river. Once across, you can rent a bike or a tuk-tuk for the day.