As for hotels, there are so many to choose from. I like to use booking.com as I can see their map of where the hotel is, reviews, and exactly what is offered. We have used that in Iceland many times and so far it has always been great.
That said, three hotels that we have stayed at in Reykjavik include Hotel Fron (closest to the center), 4th Floor Hotel and Center Hotel Laugarvatn. These last two are directly across the street from each other. I would happily return to any of these, FWIW.
As for transportation, the Flybus is very easy to use and is inexpensive. A taxi will run your about $120 US dollars. On our last trip this winter, we used the Flybus upon arrival but took a taxi for our return trip and this is what it cost us.
As for day tours, there are lots of options. Look to Reykjavik Excursions or Troll Adventures. We have used both. We used RE back in 2015 on our first trip which was a quick 3 day stopover. We did the Golden Circle with them, and it was a great introduction to the country. There are some GC tours that include Fridheimer (the tomato restaurant) as well as the Secret Lagoon. I would try to get a tour that includes these. Then this winter on our 3rd trip, we used Troll and did a one day south coast trip. It was supposed to be a 2 day/1 night trip, but a storm was rolling in so they had to cut it to one day. It was a long day but still very enjoyable.
If it works for your day of arrival, look at the Citywalk "free" walking tour. You can book it about a week in advance and they do fill up quickly, but it is a great walking tour. We have done that twice on arrival days. This winter we did a food tour on arrival day and absolutely loved that! It was also a great way to learn about the country and to try so many foods. We did the tour at 11am and there was enough food for both lunch and dinner, so in essence it was very cost effective. We used Wakeup Reykjavik for that.