Richard,
Years ago my girlfriend and I were in in Rey for 4 days (in Oct) here a few things we did. Almost everything we did, we did through (two?) local tour companies.
Quickly…
-Lava tunnel tour/hiking - 1/2day
-Golden Circle tour - full day.
This includes the main sights you read about. Going to the fault line where the Iceland is growing from. Horseback riding. Icelandic horse can do 4 gaits unlike horses from the rest of the world, due to their purebred linage. Stopping at one of the bigger, spectacular waterfalls, seeing geysers and I feel like I’m missing another stop.
-Ice Climbing on a glacier - full day due to distance. This included a stop at one of the green houses that grows/grew tomatoes.
-Northern lights - forget this one wrong time of year.
-Walking through the city. Did this on our own. Checked out some of the city artwork and buildings. Very walk-able city.
-Stop at the Blue Lagoon hot baths. It’s on the way to the airport so an easy stop. The BL is usually packed as it’s ‘the tourist stop’. We lucked out as we went off peak and it was when the world’s economy just crashed so no one was there. Otherwise we were told stop in one of the local baths. Few people and you are with the locals.
-Had whale. Not worth it. Tastes like liver. You could also try Puffin and their delicacy of rotten shark.
I think that covers it.
It was a fun vacation. Enjoy your time there and in Helsinki.
-Peter