The centre of Reykjavik is small enough to walk in twenty minutes end to end, and most of what you will want to do — Hallgrímskirkja, Harpa, the harbour, Tjörnin, the main bars and shops — is within that radius. Wear something windproof; the weather changes constantly.
For anything further out, the city bus network (Strætó) is reliable and easy. Buy the Klappið app or pay 670 ISK in cash on board (drivers do not give change). A single ticket includes a 75-minute transfer. Taxis are plentiful and metered but expensive, and Uber does not operate. If you are heading out to the countryside, Reykjavik has plenty of car-rental desks at the airport and a few in town.
The Flybus from Keflavík airport meets every flight and takes around 45 minutes to the central BSÍ terminal.