While package tours to Ayutthaya remain quite popular, it is actually quite easy to get there and get around on your own. Here's a short guide to how it's done:
Getting there: While the bus is a slightly faster and more frequent option, I prefer the train. You can get the subway directly to Hualumpong station, and the train station in Ayutthaya is very conveniently located. If you're pressed for time on the return, you can take an air-conditioned bus back.
Trains leave Hualumpong more or less hourly. There's no reason to reserve in advance. Buy a third class ticket as these trains are the most frequent. Third class doesn't have air conditioning but that's usually not a problem. The trip takes an hour to hour and a half, depending on the number of stops the train makes.
On reaching Ayutthaya, you'll need to decide how you're going to get around. Your two basic options are to engage a tuk-tuk and driver to take you around, or rent a bicycle and strike out on your own. If you want to go the tuk-tuk route, you'll find several of them waiting for you at the train station. Expect to pay 500 Baht (13.37 USD) more or less for a day. Be sure to negotiate the amount with the driver before you agree to engage him. The tuk-tuk driver will most likely have their 'standard' tour, but have an idea of what you want to see.
If you want to go with bicycles, there are plenty of those for rent near the train station, but I suggest you take the ferry across the river and rent the bicycles there, which saves you getting the bikes across the river. There are several shops renting bikes, so shop around for a bike that 'fits' well. Bikes rent for around 50 Baht (1.34 USD) per day. Of course, if you go the bike route, you'll need an idea of what you want to see. A basic minimum route would be:
You can also visit Wat Mahathat, next to Wat Ratchaburana if desired. The Wat Mahathat and Wat Ratchaburana area is a good place for lunch.
If you want to visit Chao Sam Phraya Museum, you can add it to the above first. Visiting the Chan Kasem Museum would best be done last as it's close to the east side of town where the train station is.
A more ambitious itinerary would start at Wat Chai Wattanaram and then proceed as above.
You probably want to head back to Bangkok no later than 3:00 pm to avoid the crowds on the train.