Bodh Gaya, where the Buddha attained enlightenment under the Bodhi tree, is the most worthwhile single trip from Patna. It is about 100 km south and takes 2-3 hours by train or hired car; many visitors do it in a day, though staying overnight allows time to see the modern temples built by the Sri Lankan, Thai, and Japanese sanghas.
The ruins of Nalanda, the great Mahavihara that was the world's first residential university, are 90 km southeast, easily combined with Bodh Gaya or visited separately. Vaishali, 50 km north across the Ganges, is where the Buddha gave his last sermon and where the Second Buddhist Council was held; the Ashoka pillar there is one of only two still standing on its original site.