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.
Arik, Virgin Nigeria and Aero Contractors have good scheduled domestic connections with modern aircraft to most significant destinations at reasonable prices. Their websites are very user friendly and well updated.
Note that in Lagos, the two domestic terminals, while next to each other, are about 4-5km of road which would not be wise to walk if you don't know the place from the international terminal, and you would therefore need a taxi to get from the one to the other, should you wish to transfer from an international flight to a domestic one.
Driving in Nigeria especially Lagos is somewhat unique, vaguely resembling driving in Cairo. If mastered, you should however be able to cope in most other countries on the planet. Or any other planet. Roads are bad. Expect potholes of every size. Expect people to drive on the wrong side to avoid potholes or just bad patches of road. Even on the highway. Expect the road to be gone. Expect everything. Grass or branches on the road means there is a broken down vehicule ahead of you, be careful.
Driving in Abuja, however, is relatively easier because the roads are mostly good, but since the population is lower compared to Lagos, you should adopt defensive driving especially on weekends where drunk and rich speedsters feel that there's no one on the road.
If you are white, get used to Nigerians shouting at you as you pass by. It will be something like "Oyibo", "MBakara", "Bature" or "white man" if you're white. It all means the same, they are just telling you.
Self-driving for short-term visitors unfamiliar with the roads, especially in Lagos, is by no means advisable and could actually be quite foolish if not dangerous. With crime on the rise, you could easily wander into an area or a road block set by local gangs. If you choose to rent a car, it will come with a driver familiar with the area and style of driving, which is a much easier and safer option.
If you as a foreigner wish to drive yourself, it is advisable to stick to the rules, as you will be an easy target for poorly paid police officers looking for somebody to "fine" payable directly to the officer in cash without a ticket or receipt for the most petty reasons like not indicating your intention of wanting to drive straight. Should you be pulled over, do not give them your licence, as you will then lose all bargaining power when negotiating the fine, which could easily be a maximum of all the visible cash you have on you at the time. Rather carry a copy of the licence and hand that over, or show your licence through your window. Also do not let the police get into your car. They are not really dangerous, but it could get expensive and certainly annoying.
However, if you just don't pay and never get angry, it only costs time. They have no real power over you.
Especially over weekends and festive times, it is common practice for police, especially in the richer areas of Lagos, to flag you down and wish you happy weekend/holiday/Christmas/Easter/sunny weather/trip to work. In this case, you did nothing wrong and they do not intend to "fine" you, but are rather begging for some small money for them. If you insistently yet politely refuse to give something, they will eventually let you go. Just wish them a nice weekend/holiday/etc. too.
If you work for a big company in Nigeria, you will usually have a company driver to drive you around, thereby avoiding the above mentioned problems to a large extent. He can arrange a local driving licence for you should the need arise without a driving test or proof of foreign licence.
Nigeria is not part of the most standard international Road Traffic Convention and as such will require a special International Driving Permit valid only for driving in Nigeria, Somalia and Iraq if you do not want to get the Nigerian license, not the normal one applicable to almost all other countries in the world.
Lots of street sellers surround the car when you get to crossroads in crowded areas. You should not have a problem if you keep the windows and doors locked however.
The last Saturday of the month is Sanitation Day in Lagos, when the locals clean their premises. While it is not illegal to be out on the street 07:00-10:00, due to the higher than usual presence of police officers and road checkpoints, most Nigerians choose to restrict their movements until after 10:00. Should you be caught at this time, you may be taken away by the police to perform some "public sanitation" duty, like mowing lawns, etc.