We've been as a family of five in summer and winter.
We stayed in a cottage on the golden circle the first time. Amazing volcano views and went to geyser a few times before anyone else was there and gulfoss was amazing. The restaurant at geysir hotel do a wonderful hot/ cold multicourse buffet lunch with a massive range of icelandic dishes some of which it was nice to be able to try just a tiny mouthful of. It includes tea and coffee within the price and was worth the under £20/ head.
I liked the gardens at Heveregardi because the children cooked eggs in a stream in a fishing net and then we sat at tables and ate them - maybe not an adult thing.
In Reykjavik there are numerous thermal swimming baths. If you enjoy a plunge they're great. You shower in Iceland before putting your costume on. Far cheaper than the tourist spas.
At Christmas we stayed in a cottage right on the front in Reykjavik 500m from the centre. The garden ended at the sea wall. Again we had a hot tub that had plenty of use (it helps DH's psoriasis).
There is a Reykjavik card that might be worth investing in if you're planning on doing some of the museums and using public transport as busses and leisure facilities are also included.
Some of the restaurants on the front do early diner deals. We had a fish taster platter, chips and a beer for £7.50 (December just gone). It was the best fish and chips I've ever tasted. Really light crispy batter that disappeared to moist fish.
Whale watching was fun. Some of the boats have a small cafe on board - good for a warming hot drink. They all seam to provide overalls if you want them. Theres a whale museum in Reykjavik too that we enjoyed. Not big (rather like two big sheds) but they've something like 30 full scale whale models that you can get up close too. It really helps to scale the different species.
We've hired cars each time as with such a tiny population getting out of the capital on public transport can be a little challenging. Its a very easy country to drive in and the roads are generally not very busy at all. Even the centre of Reykjavik is not difficult to negotiate - DH drove and he's a rural Welshman not a city driver. But there are daily tours from tourist hot spots.
We've not done the Blue lagoon. I think its having lots of work done at present and hear its a bit of a building site.
We have enjoyed the thermal spa at laugarvatn fontana. Lots of different small pools to wonder between. You can paddle in a thermal lake or have a steam in one of the steam rooms.
The changing rooms were lovely with nice dressing room type tables, hairdryers, cotton buds etc.
In Reykjavik there's an Ikea. We appear to be on a family mission to visit Ikea in as many countries as possible. As a family it was very convenient to get some very reasonably priced meals that we knew the children would enjoy. It wasn't all the same foods as the UK and they had some lovely deserts.
We haven't been up on the glacier, would have quite liked to do a husky sledge tour and the inside the volcano that others have mentioned - next time!