Although some transport networks in Africa are not as developed as elsewhere in the world, you can almost always find a way to get where you're going. It might not always be cheap, or comfortable, or quick, but you'll make it to your destination in one piece sooner or later.
If you're short on time, hopping on a plane to traverse this vast continent can be a great - if not terribly cheap - option. There are many airlines that will fly between African nations, usually to major cities, as well as domestic airlines that will take you from one end of a country to another.
Exercise caution when using smaller domestic carriers, especially in countries with dubious political stability, as they may not meet safety standards. These can be checked with the EU Commission on Air Safety.
Rail travel in Africa is still very much developing, improving year after year. Generally speaking, passenger railways in Africa stay within one country only, with some exceptions. There's an interconnected railway network between Botswana, Zimbabwe, Zambia, Tanzania, and Rwanda, making it a good option for travelling between those countries.
Elsewhere the quality varies. Many countries like Morocco, South Africa, and Egypt have extensive networks for passengers between major cities. Even in countries like Kenya and Botswana, which have limited routes, trains can be worth getting as a means of wildlife spotting and to enjoy the scenery. But it's likely they'll be a link in a more complicated journey, rather than your soul source of travel.
If you're looking to travel extensively overland, buses will likely become your best friends. Almost every African country has a bus service that acts as the main form of transportation for tourists and locals alike. The price and quality of these vary from country to country, so it's definitely worth doing some research before you set off. In many countries missing a bus can mean a long wait until the next one.
If an African country has a river, you can guarantee that you'll be able to travel on it. In many areas of central Africa boats are the main form of transport thanks to extensive river networks.
Along the Congo River there are old ferries that connect cities in the Congo, DR Congo, and Central African Republic. Conditions on these aren't great, so don't expect cleanliness and comfort.