I went to Iceland in Sept two years ago with my father. We spent 2 weeks there, drove the entire ring road (took about 10 days to do the ring road). We started in Reykjvaik, drove up north, east, south and then west to return back to the capital city.
Blue Lagoon and Golden Triangle were our last stops. Personally, I think Blue Lagoon and Golden Triangle are overrated, especially after visited the more spectacular sites like Myvat (hot spring), Landmannalugar (hiking + hot spring), Vatnajökull National Park (ice field, where the movie The Martian was filmed), Jökulsárlón, etc. The list goes on....
"where to go to see the best of Iceland has to offer?"
Depends what you are after? hot springs, hiking, waterfalls, ice caves, riding Icelandic horses, turf houses. To answer your question, I would go outside of Blue Lagoon and Golden Triangle to see what Iceland has to offer... You can rely on public transportation (or tour buses) to see Blue Lagoon and Golden Triangle. If you want to venture out of these 2, then definitely you need a car. Driving in Iceland is very easy, not intimidating at all.
Correct, airfare is very cheap but lodging, food and rental car are very expensive there.
It took me few months to plan out our itinerary, including booking hotels, and signed up tours. I also spent lots of time on Trip Advisor Iceland Forum to ask questions and reading other people's threads.
Hope the info above helps.