Iceland Weather in May or June

We were in Iceland in June 2012, and again in December 2013. We LOVED it! The weather in June can be cool, especially at night. We live near Seattle, and the weather in Iceland was similar to ours - days are mild (50-65F), can be rainy, nights are cooler. Check a weather website for average temps. I don't recall being too cold or too hot, but there was always wind. Wear layers and be sure to bring a wind and water proof jacket. You'll want to get close to some of the spectacular waterfalls. You can even walk through the (very wet) mist behind one or two of them. We arrived by Iceland Air in the early morning. If you do, my advice is to go immediately to the Blue Lagoon, on the way in from Keflavik Airport; it's a good way to start adapting to the local time and you save a lot of commuting time, as Reykjavik is an hour ride from Keflavik and 40 minutes from the Blue Lagoon. They have a luggage station for stowing luggage safely. In the city, start out with the Great Circle Bus Tour to get oriented and learn some history and context. Stay close in to the historical center, near the harbor so you can get most places by walking. Even if you pay a little extra, you'll save $$ in taxi fares. There is also a good bus system, though we did not use it so I can't comment on it. In the summer we rented a car and drove east along the coast as far as we could on a one-day trip. If you are there for a week, take time to fly to Akureyri on the north coast, and take a guided tour of the outlying areas for a day. Saga Tours and other agencies can line them up for you. Not cheap, but the landscape and personal tour guides were worth it for us (we did that in December). I'll try to answer other questions if you have any.

We're going to be in Iceland for just a couple days en route to Britain. We're arriving early in the morning in late May and will be picking up a rental car and driving as far as Vik. The next day we'll head back and see the sites we missed and make sure we try some of the "famous" Icelandic hot dogs before spending the night at a B&B near the airport for our early morning flight out.

We want to make sure that we see all we can in our short stay - can you provide us with recommendations on the "not to miss" stops?