My wife and I went to Reykjavik in October last year for a few days. We didn't hire a car, we got the bus from Keflavik airport, it stops at various points on the way and at lots of places in Reykjavik itself. We stayed at the Sand Hotel on Laugavegur - it was nice, centrally located, but quite expensive. Round the hotel is the main touristy area so there are lots of things to see in walking distance. There are a few other hotels in that area, lots more, including the hotel chains, are a little further out towards the football stadium.
We did one of the bus tours which goes round the Golden Circle taking in national parks, geysirs, Gullfoss waterfalls, the secret lagoon. I can't remember the name of the company we used, but there lots which do it all with variations on what they visit. There are other tours to other places nearby as well.
In Reykjavik itself there's lots to see, the cathedral, maritime museum, Harpa concert hall, Sun Voyager, phallalogical museum. Next to the football ground is Laugardaslaug, a huge heated outdoor swimming pool complex, with lots of thermal pools round it. There are loads of places to eat and drink, a few really good bakeries and cafes as well. At the cheap end is the Chuck Norris grill - wings, ribs, etc, no more expensive for that sort of thing than here. At the other end of the scale, I'd recommend Meat, it was fantastic, but well over £100 a head for dinner. Lots of pubs so it depends on your taste, we liked Den Danske Kro, traditional pub, decent beer, live music from local singers. Bastard Brew is a modern microbrewery, food was okay, beer not bad, but poor service, hipster central if that's your thing.
If your return flight is leaving early as ours was, I'd recommend the Hotel Berg in Reykjanesbaer. It's modern, clean, comfortable and reasonably priced and they'll taxi you to the airport at any time you like, it's only 5 minutes away. You can get a bus from Reykjavik to Reykjanesbaer, it's a few minutes walk from the stop to the hotel.
Enjoy your trip, it's a great place to visit.