If you want to ride the horses, I remember there were a few riding schools close to Reykjavik that offered (half)day trips, so it's definately possible! We thought it was a little expensive, but if you're a big fan of horses it it might be worth it
If you want to see more of the countryside you could do the 'golden circle' either with an organized tour or with a rental car. (It goes to the geysir, the biggest waterfall (gullfoss) and you can walk between the european and american tectonic plates. Our tour also went to a black sand beach and we fed some horses on our way back, but I think that's not included in most tours). Actually, if you want some freedom, i'd recommend hiring a car for 24hours. You'll be able to see more that way.. oh and the yoghurt is called skyr and it's absolutely amazing!
If you want to see more of the countryside you could do the 'golden circle' either with an organized tour or with a rental car. (It goes to the geysir, the biggest waterfall (gullfoss) and you can walk between the european and american tectonic plates. Our tour also went to a black sand beach and we fed some horses on our way back, but I think that's not included in most tours). Actually, if you want some freedom, i'd recommend hiring a car for 24hours. You'll be able to see more that way.. oh and the yoghurt is called skyr and it's absolutely amazing!