We spent 2 nights in Reykjavik and 2 nights in Skogar in June, and found Skogar to be a great base for our Southeastern adventures. I'd highly recommend doing a glacier hike or ice climbing. That would be a 3 hour drive east from Skogar, and we went with Glacier Adventures for a ~4 hour tour. Best day of the entire trip, and you won't regret it. Also stopped at Jokulsarlon Lagoon on the way back home that night, which is great for photo ops. We stayed at the Skogar Guest House here, which was lovely and a 10 minute walk to Skogafoss, the gigantic waterfall you see on every Iceland pic.
In Reykjavik, we did the Into the Volcano tour, a half day tour that takes you into a dormant volcano. Again, highly recommend and so different from anything I'd done before. You get a taste of very different landscapes every few hundred km in Iceland, so take your time driving from place to place. Stayed at the Radisson Blu literally right next to the famous hot dog stand in Reykjavik, which was a great central location.