We went a few years ago in late June and the sun never set!
How much time do you have? We went all the way to the far northwest corner and hung out there for a few days. It is wonderful and home to the Latrabjar Cliffs where there are the most puffins and other birds. We also drove to Geyser, Thingvellir, Gulfoss, etc., and those are all really easy to get to from Reykyavik and easy to see. No 30 mile entrance roads like you would have at our National parks.
We skipped Blue Lagoon. It is expensive, there was a line and we had a long trip ahead. We did find another pool on the side of the road out in the country and just changed in the car and hopped in. The 7 public pools in Reykjavik are supposed to be great too - each one is different.
In Reykavik we loved Willlie Knudsen’s Volcano movie and the Saturday evening car parade along the main drag - I think it’s called the Laungevagar or something.
We didn’t go to the Snaefellnes peninsula but it seems there is always snow there. It looked beautiful from a distance.