Papeete is a walking city. It's really too small to bother with any other form of transport, unless you are going out to the fringes, or would simply like to experience the famous le truck for fun hop-on, hop-off, anywhere in the city center for about XPF100! Bring a water bottle: it can be quite hot and humid.
Don't bother with taxis: they're extremely expensive and very hard to find after 6pm, apart from two dedicated taxi stands along the waterfront. Meters are unheard of, so be sure to confirm the fare in French, if possible before getting into a taxi, and don't be afraid to protest or refuse to ride if you think the fare too high; as a general rule, you should never have to pay more than XPF 1500 for a journey from one side of the city center to the other. Many drivers distribute calling cards when you disembark; if you'll be relying on taxi transport for whatever reason during your stay, it's definitely worth becoming a repeat customer with a driver you trust and who will give you a good deal.
Le Truck will take you to other parts of the island and around town quite cheaply.