I went in May with two friends, none of us had been there before. It ended up really easy to get around so have no fear! We rented a 4WD and only needed it in a few places - although in May the interior roads were all still closed. With that in mind we pretty much stuck to the ring road. We booked hotels in advance, which had the benefit of peace of mind (there was always a place to sleep) but it did limit our flexibility. Most of the "hotels" were rooms in houses and included a European breakfast of meat, cheese, bread and coffee. The downside is that this limited our flexibility to chase the light. If I went back (particularly with only one other person) I'd be really tempted to make use of a camper van for at least part of the trip.
So that said, here's how we structured our trip (which included a lot of driving). You can look at my instagram (@imageskylejones) for photo examples until I get some planned blog posts written up.
We started in Myvatn in the North - and we didn't have enough time there. The lake itself is cool with some thermal and volcanic areas around it. We hit Godafoss (a must see in the North), Dettifoss (foss=waterfall), Selfoss (I really want to see this one again in better conditions), and the Grjotagja cave (rumored to be used in Game of Thrones but I don't believe it). Lot's of other things we could have done but didn't make it to.
Our next home base was a hotel in Gullfoss which is part of the golden circle (high traffic tourist area). From there we hit Gullfoss itself (I was more impressed than I'd expected), Geysir (like a mini Yellowstone), and Bruarfoss (I'll post one from there today). We also made a day trip to Kirkjufell which is a must see. I was already a little burnt out from the hundreds of photos I've seen of that pointy mountain and waterfall - but you have to make your pilgrimage. The rest of the Snaefellsness peninsula has some really cool stuff too.
We spent the most nights in Hofn on the south east coast. That allowed a day/night trip to the Jokulsarlon ice lagoon and beach (most fun morning shoot I've ever had) as well as the Glaciers (we did a cool Glacier hike/tour) and the Skaftafell national park. Around Hofn itself is the Vestrahorn (mountain rising up from a black sand beach) which I wanted to shoot multiple times but the weather didn't cooperate. We also took a trip up the east coast which could use more attention as well. Our main goal was Borgarfjörður eystri and the puffin colony.
We didn't spend much time around the South - with bad weather in Vik - but that has the famous black sand beach and the famous Skogafoss and Seljalandsfoss waterfalls nearby. Since you have more days than we did, you can do some of it more justice.
But here is my main photographic recommendation. Plan your schedule to shoot through sunset and sunrise. The light during the "night" is amazing and most of the people are gone. You can sleep during the day.
Let me know if you'd like any specifics. I can share pictures with details.