Iceland is amazing, but possibly not if you're not outdoorsy or into museums, that's kind of what it does best. The landscapes are just other worldly.
There isn't one single time of year that's best to visit because it's so different throughout the year, winter for northern lights, summer for long days, autumn, I think for whale watching. If you're not used to it, you might find all the outdoor bathing in winter too cold (the water is warm/hot but obviously it could be icy cold/snowing). Plus not everything is accessible all year round, eg I wanted to visit Videy but the ferries are seasonal so not running when we went.
In Reykjavik, we liked the pond, waterfront, Perlan museum, big relief map of Iceland in the town hall and just wandering around. We also went on the Golden Circle tour and whale watching. The map in the town hall illustrated how, while it felt that we'd seen loads, we'd only actually travelled round a tiny corner of the country, it's not somewhere you can tick off as having 'done' in a single mini break, like you wouldn't think that if you've spent a few days in London that you've experienced the whole of England.
We booked the tours locally, it was all very easy and well organised. A lot of them had a guarantee where if you didn't see the northern lights/any whales, you could go again for free, so possibly don't do what we did and book the trip the day before you left - we didn't see any whales, but couldn't go again as we were leaving for the airport first thing next day.