Are there specific tour groups that were good to use? There are many listed and I don't know which is better. Should we consider renting a car?
We rented a car and did the ring road around the island in 10 days. Icelandair has a good package that includes airfare, hotels (and breakfast), car rental with Garmin, Flybus transfers to and from the airport, SIM card for your cell phone. We got a good overview of the island. The Icelandic people are wonderful - polite and very accommodating. We want to go back as there is so much more to see and do in Iceland. Ten days was not enough. If you are staying in Reykjavik, there are days tours that you can take such as pony trekking, glacier hikes, Golden Circle tours, whale watching, puffin cruises (seasonal). The tours pick you up and drop you off at your hotel. Iceland is amazing. You will love it.
It depends on your length of stay, which part of the country you wish to see, and just personal preference of whether or not you like having a guide. I have been on a tour though Icelandic Saga which I greatly enjoyed. But it is very easy to get around the country with a car.
Iceland or otherwise, I always recommend a guide for at least a day trip -- there's a lot of culture and local nuance that a traveler misses when they see something on their own and are only going off of a book or web site.
I went in winter to see the Northern Lights, so I joined a tour group as I was traveling solo and did not want to drive alone in the snow. I booked mine with Extreme Iceland (I think lol), but the tour ended up being run by this tour group. They were a good tour company.