YES!! I used to live in Iceland! It’s my absolute favorite place in the whole world and you should take as much time as you can get off, because there are TONS of things to do in the summer. I’ll make you a list of things I loved to do when I lived there, and also things my family enjoyed when they visited.
Blue Lagoon-about 15 minutes from the airport on your way into Reykjavik. This was my parents and siblings favorite part of the entire trip and it’s a MUST. You’ll want to block off the better part of the day for it, because you’ll never want to leave. Make sure you use the silica mud-your skin will thank you for it.
In Reykjavik, there are a bunch of great shops and bars on Laugavegur. A place called “The Deli” has the best pizza in town, Bæjarins Beztu is this tiny little hot dog shop that Icelanders are very proud of since Bill Clinton ate there, a bar called Hresso has a wonderful fish soup (definitely try the fish and lamb in Iceland if you can, it’s the best!), kjottsupa is lamb soup and my family had it about 10 times while they visited. Cafe Rot has great coffee and is a fun hangout, also, Vegamot and Argentina are really wonderful restaurants if you want a recommendation on that.
Two streets off Laugavegur is the harbor, and you can find whale watching tours and they’ll take you to puffin island-I loved both of these.
The Viking Museum is very good, lots of old artifacts and you get a good sense of the history.
The Golden Circle Tour is the most popular as PP said, and it’ll take you to the major sites, Geysir, Gullfoss, Thingvellir (the national park) all these are great 🙂
If you like waterfalls, I’d check out Seljalandsfoss which is one of the most famous waterfalls in Iceland (stunning!)-you can walk behind it and climb to the top.
Husafell is a town about an hour and a half outside of Reykjavik, you can see Barnafoss and Hraunfossar (waterfalls) and also it’s really close to Langjokull Glacier, which you can take a tour on or just go see and explore yourselves. There are also some underground caves in the area Víðgelmir and Surtshellir.
Hekla the volcano is still erupting I believe, and they do do tours and take you see them, so that could be a once in a lifetime opportunity if you can get up the courage!
Esjan is the mountain in Reykjavik, and you can climb it and put your name in the book at the top, I know a lot of tourists that do this.
Krysuvik is a good tourist spot as well, it’s the the geothermal fields and hot springs. Hot springs are a big part of the energy system in Iceland, so it’s worth a look I think.
There are horseback riding tours that will take you along the mountains and countryside if you like that sort of thing.
Hopefully this will get you started 🙂