My suggestion: take a package holiday in a four wheel drive coach in which the
overnight accomodation is either tents or the camping-huts.
The minimum reasonable time to go from Reykjavik, round the south of the island,the east of the island, then back across the centre is 2 weeks, and 3 weeks is
preferable.
Try and see: Thingvellir, Thorsmork, Skaftafell, Asbergi/Jokulskarlon, Myvatn