There is a perfect hike just outside of Reykjavik and it is even accessible by public transportation. From the city you can see the mountain range called Mount Esja and one nice place up there is the Pverfellshorn. To go there you can take the bus 57 to Esjustofa Hiking Center. There is a nice cafe at the beginning of the trail. Then the path leads along a river up to a place called Steinn. The view is amazing! And it gets better the higher you come. At Steinn the path becomes more steep and in the end you have to climb a bit, there are also some fixed ropes. I stayed about two hours on the top and enjoyed the perfect view before walking the same way down 🙂