I just got back from a seven day trip to Iceland. However, it was my second time there so we did skip over a few things. Also, it was mostly a driving trip with small hikes at most of our stops. Although hiking is wonderful, we chose to drive for most of the time to see as much as possible. I do plan on going back and I will probably hike a lot more.
Our itinerary was as follows:
Day 1: Bus from Keflavik to the Blue Lagoon then to Reykjavik>
Day 2: Full day in Reykjavik
Day 3: Rented a 4x4 and toured the Golden Circle (Thingvellir, Geysir, Gullfoss) and then camped in Landmannalaugar.
Day 4: Explored (relatively briefly) Landmannalaugar and Eldja. Seljalandsfoss, Skogafoss and Vík. We had to crash in Vík at about 9pm because we only had about 2 hours of sleep since we had left Reykjavik.
Day5: Laki, Skaftafell, Jökulsálron and arrived at Egilstaddir at 2:30am.
Day 6: Askja, Dettifoss, Asbyrgi, Godafoss, Myvatn(briefly) and Dimmuborgir. Arrived at Akureyri well after midnight.
Day 7: Akureyri to Keflavik.
While I am happy and satisfied with what we saw and did, we could have really used an extra week and probably more, to spend more time at all our stops and also to explore the Westfjords, but we were on a tight budget. Keep in mind though, that out itinerary was extremely packed. We only slept for 2 of the 5 nights on the road and we never had a hike longer than 2 hours.
I would recommend hiking where possible, but I believe a car, more specifically a 4x4, is necessary to see a lot the diverse natural beauty that Iceland has to offer.