Toward the bottom of Bankastraeti (town centre) there is a tourist information office in a small lane where you can get answers to more or less any question and advice on what to see and do. There are some museums worth visiting, particularly the National Museum and the National Library (saga manuscripts) if you want to see how culturally advanced the early settlers of Iceland already were 800-1000 years ago. If you are interested in learning how the society operates today, spending time to discover some of the ways and reasons why Icelanders come tops in all sorts of statistics (infant survival, life expectancy, literacy, per capita income, non-existence of real poverty, etc.), how they top polls of "satisfaction/happiness", etc., etc. is worth doing if only to compare with other societies, including your own. There is a trail that goes completely around Reykjavik which is pleasant to walk (see local maps and signs), especially in the middle of a summer's night when the weather is good. Best divide it at least into two sections. There are also several swimming pools and "hot pot" leisure facilities (geothermal water) which are a pleasant experience, but note that hygiene has a higher value in Iceland than in the UK and you are required to wash properly (i.e. naked, in segregated changing rooms) before going in. There are lots of different cafes restaurants in Reykjavik, seafood being particularly good due to the very fresh and high quality raw materials. Some are quite expensive, some are moderately priced - ask anyone for advice, people are quite ready to be approached, unlike in some countries where it is considered extremely rude to turn to a stranger when needing help or advice, never be afraid to stop anyone anywhere. Note that alcoholic drinks are universally expensive due to the taxation policy and the view that alcohol is not a life necessity - nobody is likely to bat an eyelid if you stick to non-alcoholic (unless perhaps you happen to be at a bar).