First of all you should visit Reykjavík city. Reykjavík is probably one of most beautiful cities in the world. It has excellent restaurants, hotels and shops. You should also visit Reykjavík center (the old town) there can you enjoy real Icelandic culture.
Thingvellir is the national park of Iceland. They are visited by thousands of tourist every year. There can you find hiking trails, camping ground and you can go fishing and the nature there is beautiful.
Gullfoss and Geysir are places you must see. These places are visited by thousands of tourist every year like Thingvellir. There is also hotels and shops.