Ayutthaya is quite an expanded city and it will hard for you to entierly visit it by feet. So you should probably rent a songthaew for an afternoon to see all the different interesting point of view. There ar mostly in the center of the city but can be far from each other. Watch out for beaters, they will wait for you as soon as you leave the bus for proposing you to take a city tour. Their prices are very high and they will show the city from a taxi without any guide. If you want to visit everything, it should takes you one or two very long days. You can almost choose to take a cruise that will show you most of the main temples.
3 things to see in Ayutthaya
Wat Panan Choeng (Golden Buddha)
With 19 meters high, that golden buddha is massive. The room in which it is quite narrow and you might find difficult to getting in without breaking the serenity of the place. But it is an ummissable of the city. The temple itself is very enjoyable, there are a lot of offering places (golden leef to put on statues,encent, flower or also foods). Go out by the right of the temple and get to the river to see the fishes
Wat Chai Watthanaram
This temple is not as old as it looks. He is dating from the Kmers invasion in from the begining of the 20th century. You can choose to have a walk in the middle of thedifferents parts of the place. Please be respectfull, it is still a cult place. As you may see there are still many monks there. The small rock slab are only for meditation. You can also do an offering the the buddha with a big piece of clothe, a local perso will help you put it in place.
Wat Yai Chai Mongkol
This site is one the most famous in Thaïland with his iconic Buddha’s head in a tree. The place is not as crowded as you could believe. And the entrance is only 50 Baths, not expensive for what it is. The place is quite big and there are many ruins all over the place. You can easly spot a temple shape, in the middle of what was before a city. If you are there at the closing time, the market is very near and you will find everything you need to have a very good meal.