Ayutthaya trip guide, the best is to start at Bangkok's Hua Lamphong train station. Forget all other options since you will just get stuck in the gridlock traffic and spend half of your time there. Hop on the train to the north and have a interesting view into the rural live in Thailand.
Since you probably are first time in the city the best is before you take a ride walk to the nearby Ayudhya Riverside Hotel and ask for a map they have a very good and simple map. If you are hungry now or later use their floating restaurant with a splendid view over the river and the ships cruising. An alternative would be to tell the motorbike guy to bring you to the Tourist Information Center.
To get a right orientation do an Ayutthaya cruise as a starter, the long tail boats are waiting at different locations. The best is tell the motorbike guy to get you to Chantharakasem National Museum, which is only about 5 minutes ride, now tour this small museum. The place give a good insight into the living of royalty at that time and plenty of items are on display including old Buddha artifacts, ancient weapons and in the center of the place is a watchtower with a excellent view over the area, the river and the channels, the historical park show more temples and artifacts than the historical park at Sukhothai.