I would recommend that you plan to stay overnight in Agra. This gives you more time and comfort. The places I would recommend in Agra are the Fort, Taj Mahal. Nearby is the other capital: Fatehpur Sikri - worth seeing.
There are several hotels in walking distance from the Taj. I believe that either the Oberoi group or the Taj group has one facing the Taj Mahal.
Regarding transport, a car would be the best option, due to extra logistics with the others. But do hire a good guide in Agra.