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Photo: Helgi Halldórsson from Reykjavík, Iceland, CC BY-SA 2.0 via Wikimedia Commons
Icelanders are proud of their LGBT community, and every August they show it with one of the biggest annual festivals in Reykjavík. Typically a parade will wind its way through the city with floats of varying degrees of outrageousness. It then ends at Arnarhóll with a large outdoors concert. Gay bars and bars that normally don't self-identify as gay tend to be very full this evening. In the preceding days there are various events celebrating LGBT culture.
Reykjavík Pride is Iceland's largest human rights festival, focused on LGBT+ issues. The festival has been held annually in Reykjavík since 1999.
The Pond
A small lake in the centre of the city where young and old often gather to feed the ducks. The Icelandic name, Tjörnin, literally means "The Pond". Tjörnin is mostly surrounded by a park called Hljómskálagarðurinn (Music Pavilion Park) which gets very popular in good weather. The southern end of Tjörnin links it to the Vatnsmýri swamp, a small bird reserve with paths open to the public except during egg hatching season. Built into Tjörnin on the northern side is Reykjavík City Hall.
A small park (or square, depending on definitions) in the heart of Reykjavík. It's many locals' favorite place to spend sunny days, either at one of the cafés lining the north of the square or simply having a picnic on the grass. The parliament and the national cathedral both stand by Austurvöllur.
Klambratún is a park just east of the city centre on an area which remained farmland while the city was built up around it. The area was later converted into one of the largest public parks in the city and often hosts various events. One of the houses of the Reykjavík Art Museum, Kjarvalsstaðir, is inside the park.
Grasagarður Reykavíkur
The Reykjavík Botanical Gardens are not large, but they're nice for a short stroll and a good place to see some of the plants that grow in Iceland.
Viðey is a large island in Kollafjörður, the fjord to the north of Reykjavík. It used to be inhabited, and in the early 20th century it had a small fishing village. Nobody lives there anymore apart from the birds, but it's a popular way to get away from the city without leaving it. During the summer, a café is operated in one of the houses on the island. The building was built for Skúli Magnússon, an 18th-century politician often called "the founder of Reykjavík" and designed by the same man as the royal palace in Copenhagen - although it is not quite of the same scale. Among its more modern architecture, Viðey is home to the Imagine Peace Tower by Yoko Ono (see below). To get to Viðey you must take a ferry from Sundahöfn, some distance from central Reykjavík (on bus route 5). The schedule and prices can be found here.
At the far western end of the peninsula on which Reykjavík sits there is a small island. This island, called Grótta, is connected to the mainland on low tides and open to the public most of the year (closed May 1-Jul 15). Just make sure you don't get stuck on the island when the tide comes in!