The international flights to Iceland are through Keflavik airport (KEF) 30 minutes drive from Reykjavik. Icelandair has a special deal on a two-day stopover at no cost for flights between Europe and America.
The easiest way to get to Reykjavik Central is by bus. It is well advertised and you will find it easily. You can either take the bus to the main bus station (BSI) and walk from there or take a shuttle bus forwarding you to your Hotel. The walk to the hotel from the main bus station is 15 minutes. The Price is probably close to 4.000 ISKR. Another alternative is a Taxi. Be sure to use a legit Taxi service, such as Hreyfill. A normal price is probably close to 25.000 ISKR.
A map of Reykjavik central. The red numbers show:
1 – (beneath Vatnsmýrarvegur) The main bus station (BSI) where the bus from Keflavik arrives.
2 – (beneath Háskóli) The Venue
3 – Hotel Odinsve
4 – Hotel Holt
To travel inside Reykjavik you can use the bus system. There is an app that shows the easiest way to get round. The link to it is here : Home – Klappið
Day tours leave from BSI but thre are pickup points at other locations. Some of them are in the city center. See here: Pick-Up Locations and Designated Bus Stops in Reykjavík | Reykjavik Excursions