I went last summer for the first time and then again this summer. I love Iceland!
I'd recommend going in the summer as the days are so long, giving you much more time to do things. The first time we stayed in an Airbnb in Reykjavik and bought the City Card which gets you unlimited travel on buses in and around Reykjavik, into the many swimming pools which are open from early morning till about 9 or 10pm, and in to museums and galleries. We did the Golden Circle tour with Greyline Coach Tours. The airport is about 50 minutes drive from Reykjavik and we got the Flybus to and from the airport, booking in advance online.
This year, I hired a car and we stayed in hostels. First and last nights in Reykjavik, then travelled up the West Coast - stayed a couple of nights in hostels there, then stayed in a couple of different locations along the South Coast. Eating out is eye-watering expensive, but they love hot dogs, which are reasonable, widely available from stands and service stations, and pretty nice. Bonus is a good supermarket for food to cook in hostels. I joined Hostelling International and got about 10% discount. Apart from the hostel in Reykavik we, my teenage DS and I, stayed in twin rooms so it's not all about sharing dorms with strangers!
Last year we flew with EasyJet and this year with Wowair which is an Icelandic budget airline. Rented car from Blue Car rentals - pick up and drop off from the airport - 5 minutes walk from terminal building. The speed limits are lower than here, and once you're out of the city, the roads are very quiet. There really is only the one major route round Iceland, the Ring Road, with other minor roads off. Some are not suitable for anything less than a 4x4, but you can get pretty much to see what you want from paved roads.
A couple of websites I like are
icelandmag.is
www.inspiredbyiceland.com/plan-your-trip
You won't regret going. It is an amazing place!