Reykjavik City and surrounding communities run a public bus network under the name 'Strætó bs'.
Check out www.straeto.is/en for detailed information about bus schedule, name of routes and prices.
You can see where the buses are in real time on a map on their front page where you can select the route numbers to monitor.
They also have a nice app for your phone and Moovit is another nice app you can use here.
More public transport
The website publictransport.is shows all different modes of public transport, bus, ferry and scheduled flights.
