Are there cheap local buses or taxis from REK into town? The web keeps showing places charging $40 USD per person each way transfer fees? None of the moderate or lower hotels seem to have free shuttles.
Keyflavik airport is quite a ways from downtown Reykjavik. Check out:
http://www.kefairport.is/English/Transportation/Buses/ You can also combine an excursion to the Blue Lagoon, with a pick-up or drop-off at the airport. We did it on our way back to the airport.
Be careful using the REK code for Keflavik airport. Reykjavik has two airports: Keflavik (KEF) for international flights which is way out of town, and Reykjavik City Airport (RKV) for domestic flights and flights to Greenland and the Faroes. Using REK might have someone thinking you mean the City Airport which is in town and close to everything.