It would be best to travel around in your own car or a hired one with a driver but there are various other modes of transport. The road systems in Nigeria are relatively poor compared with North American and European countries, but often still passable. The "okada" motorcycle is not for the faint-hearted there used to be no helmets but as a law the rider is required to have two helmets for himself and a passenger and should only be used for short distance journeys. "Okadas" will get you to where you want to go quickly and you will get there in one piece. In Lagos, there are lots of buses and taxis. There are two main types of buses, the molue and the danfo. Most smaller cities have more taxis than buses, and they are quite affordable.
For travelling from one city to another, you go to the "motor park", find the taxi that's going to your destination, and wait until it "fills up". The price is fixed, you don't have to negotiate. Some drivers may have a risky driving style however - practically this means that the only rule consistently adhered to by cars, not necessarily motorcycles, is keeping on the right.