The best way to see Iceland is to rent a car and drive Ring Road. There are a lot of day or weekend excursions that leave from Reykjavik, if that's your thing. I can't speak to it since we planned our own adventures.
PLACES TO VISIT
Gulfoss falls (golden waterfall)
Majestic waterfalls outside the city with a great cafe at its base.
On the way to Gulfoss you'll pass the geysirs, so you may as well stop. Plus it's pretty cool to watch them erupt.
Black sand beaches anyone? Vik is a small town right on the south coast. We stopped at the cutest cafe for lunch, Sudur.
Svartifoss waterfall at Skaftafell National Park
Stunning. Worth the hike up.
Jökulsárlón glacier lagoon
Icebergs! Glaciers! Oh my! Seriously a must.
Yes, it's touristy but you should do it. When else will you swim in a geothermal pool?
And the side of the road
Seriously there are so many picturesque moments along the highway that you'll want to stop and capture the breathtaking Icelandic scenery.
PLACES TO STAY
Frost & Fire Hotel (Frost og Funi)
Awesome hotel with delicious European-style breakfast (bread, cheese + cucumber is my fav).
Three of the cutest cabins at the base of these giant mountains. Oh and there was a sheep farm. 15 minutes for Skaftafell National Park where we hiked to see Svartifoss waterfall. A 30-40 minute drive to the glacier lagoon in Jökulsárlón. Stop in Vik and load up on groceries since the cabin has a fridge and stovetop.
And I forgot to mention, this is the Airbnb studio apartment we stayed in for four nights in Reykjavik. It was perfect. South of the city by the university and minutes from the coast which had an awesome paved path for biking and running. Bjorns Bakeri was a block away (which we didnt' realize until the day before we left) and it was only a 10 minute walk to the domestic airport to pick up our rental car.
If you have any questions about Reykjavik or Iceland, let me know!