Getting around Ibadan without your own transport takes patience. The city has taxis and minibuses, but most are in rough shape. Do not be surprised to find three people in the front seat and four in the back, with the driver waving you in because "there's loads of space."
The best strategy is to arrange transport in advance. If that is not possible, spend some time finding a taxi in decent condition, then negotiate with the driver for the duration of your stay. Most will be happy with the arrangement. Ibadan is circular, and the road from Lagos winds nearly halfway around the city's outskirts, so knowing your route matters.
Roads are not well maintained, and conditions worsen significantly during the rainy season. There are no warnings for dangerous curves or obstacles, so driving with limited visibility is risky. During the rains, a taxi or rental car beats walking or riding a scooter.