Central Cayenne is small enough to walk — most of the sights cluster within ten minutes of the Place des Palmistes. The streets are flat, the heat is heavy, and you will sweat through everything you wear, so plan walks for early morning or late afternoon.
For trips out to the airport, Montjoly beach or onward to Kourou and the Spaceport, taxis (TaxiCo and a handful of private operators) and rental cars are the practical choices — there is a public bus network (Agglo'Bus) but it is slow and aimed at locals. Driving is straightforward, on the right, on French-quality sealed roads.