It'll be quite easy for you to get around during your trip to Iceland. In this small country, the best option remains the car but buses also allow you to travel between the majority of cities and towns in the country.
Upon arriving in Iceland the best way to reach the city centre if you don't have a hire car waiting for you is to take the flybus to Reykjavik city centre, located 45 kilometres away. The shuttles are connected to the flight timetables and will drop you off right at the main bus station. If you choose the "flybus+" option, another small shuttle will take you directly to your accommodation.
Reykjavik is a small capital and it's very easy to get around it all on foot. If your accommodation is a bit further away or if you want to visit some places on the outskirts of the centre, the bus is the best option. There are several lines and maps are available from the tourist office. Just be aware that you must have the exact amount to buy your ticket from the bus driver.
Some domestic flights connect the capital of Iceland to other cities in the country such as Akureyri in the North, or Egilsstadir in the East. The domestic airport is in the city centre and is separate from the international airport which is located in the Reykjanes peninsula. Iceland does not have trains so the car remains the best method of transport to get around the whole island in complete freedom. A city car is fine if you're staying on the main road, route number 1. In Iceland a 4x4 is necessary in the middle of winter or in summer to go across the inland. Besides a car which allows you to get around the whole island, there are also several bus connections. The timetables and frequency are different between summer and winter. In summer there can be several departures per day and special connections are available, in particular to the inland region. In winter traffic is restrained and some lines only go every other day. Make sure to check the timetables properly.