I have toured Iceland 2 years ago for 10 days and circled the whole island (route 1) driving a rented car. As small as this is Iceland appears to be on the map we barely managed to circle it in 10 days. The "touristy" places you mentioned are lovely. We were unable to get to Landsmannalaugar. But we were told that this is an exciting trip and probably quite unique.
Therefore, we decided to visit Landsmannalaugar on our upcoming return trip to this most exciting country.
Since I find it very difficult to find 1 day return trip from Reykjavik, I will be most grateful if you could advise me the name of the company that your group are using going there. We were warned not to attempt driving there by ourselves (even not eith a 4X4 vehicle).
Have a great trip!
AA

topping

Geysir and Gullfoss are exceptional and if you can get to them on your way to where you're going by all means do it. (with a group it may not be as expensive as usual for a car) If you're going out hiking then you will not miss what you would get at Thingvellir. Landmannalaugar isn't something I know alot about, but have seen the pictures and it's probably a decent choice. I have talked with people who have been to Lake Myvatn which is what you're looking for as well. But unfortunately I can't see where you have time for that.

If it's not specifically hiking you want, and you're OK with something a little more expensive - try going to the Westman Islands. The need for a flight keeps most tourists from ever getting there, yet it's a place where you can still feel the hot lava! Very unique.

Landmannalaugar is absolutely positively GORGEOUS!! We styed there overnight and did some hinking in the area, and the scenery (the colors) were just amazing and totally different from anything else on the island. Mind you - we had great weather (by Icelandic standards), and the place tends to be well-visited (again, by Icelandic standards). Definitely take your bathing suit with you when you go.
Have fun!