Abidjan is quite spread out so walking can take a lot of time and bicycle riding isn't the safest choice except nearer the water in Zone Quatre. As such, there are many options to get around via motored transport.
By bus
There are several bus routes throughout the city. They are cheap and decently reliable, although they are often incredibly crowded due to insufficient numbers. Some of the bus stations can be overwhelming though, such as Adjame which, for those new to travel in West African cities will be a lot to handle. There is also the threat of pickpockets in these crowded areas.
Moto
In 2010, the Ivorian government relaxed import restrictions on small motorcycles. Prior to this, the amount of motorcycles you would see on the street was negligible and there were absolutely none acting as taxis as it was illegal. Times are changing on this front, but be forewarned that going about Abidjan on the back of a moto is probably the number one way to die during travels, although it is cheap.
Taxis
They have a complex system that is comprised of two types of car taxis. The first type that most visitors will encounter are the orange or red-orange ones. These are legally able to operate anywhere in the city and you will most likely be able to ride solo in them. They are also the most expensive. A ride from the airport will run most people especially non-Africans who speak little French about 5,000 CFA, even to districts that are just 3km away. If willing to haggle a lot the drivers will often complain that they have to pay a fee to pick up passengers there, which is a lie you may be able to get it to 3,500 or 2,500 CFA. A ride between two distant districts such as Zone Quatre and Plateau will be about 2,000 CFA.
The other type of taxi is color-coded to operate in a specific neighborhood, such as the green taxis you'll see in an area such as Koumassi. These are significantly cheaper, but will most likely have to be shared and of course the distance they can travel is limited to a single neighborhood.
Travel books often make allusion to some taxis having meters. If they do and this is rare, they are never working and you always, always agree on the price prior to departure.
Boat
If you just need to cross the lagoon and can make use of one of the ferry routes, by all means take it. While the lagoon is polluted in some parts, it's still a wonderful ride and gazing at the Abidjan skyline from the water at sunset is delightful.