The Reykjavík City Card is the easiest and most inexpensive way to experience Reykjavík city!
The Reykjavík City Card offers free unlimited travel by bus within the Reykjavik Capital Area (it does not include the bus nr. 55 to the airport as that leaves the Capital) and free entry to a great selection of museums and galleries and all swimming pools in Reykjavík. In addition, the card also gives discounts on various tours, in shops, and on services.
You can buy the card online at visitreykjavik.is, then you receive a voucher via email and can then pick up the City Card at certain locations, you can see the list here. Note that you can NOT use the voucher on the buses.
Here you can see more information about the City Card.