1: Getting to know Reykjavík

It is nice getting to know the city. Here is a map of Reykjavík, divided into many areas. (Area 7 and 8 show the image below).
Another map of Reykavík and its surroundings is here. Following image is from that map.

The city can be entered by car, bus, bicycle or by foot. (1) Start at Perlan (the Pearl), built on Reykjavik's hot water tanks. Go up to the fourth floor where you can go out and get a view of the whole city. On the ground floor there is an artificial geyser and outside there is another one.
(2) Drive on to 'Hallgrimskirkja', the biggest church in Iceland. You can't miss seeing the tower and it can be your guiding light in the city. Usually it is not hard finding a parking place near the church. In front of it, there is a stature of Leif the Lucky who was the first European to sail to America.
Take a walk down Skolavordustigur. If you want you can go down Laugavegur as well (the main shopping street) or if the walking is enough for the kids, you can drive down to (3) Reykjavik's pond (Tjörnin) and park in the car-parking-house under Reykjavík's City Hall by the pond (5 min drive). Look inside Reykjavik's City Hall. Inside there is a big 3D map of Iceland which is worth taking a look at. Bring with you some bread and feed the ducks, swans and geese on the pond. On the other side of the pond, there is a nice garden where it is nice to run around.
(4) Close by the pond is The Parliament House, a stone house where the parliament resides. It stands by Austurvollur, a small squere with a statue of Jon Sigurdsson, our Leader for Freedom. Iceland became independent from Denmark in 1944, on Jon's Sigurdsson's birthday, the 17th of June. 17th of June is now the National day of Iceland.
Dómkirkjan (the Dom Church) is on the other side of The Parliament House. This is as much downtown as you can go. There are some old houses, café's and restaurants and on a nice day you can sit outside and enjoy refreshments.
Maybe this is enough of 'city' for the kids and time for some activity. Drive to the Vesturbær (the West-city or west-part of the city), pass the University (5) and the National Museum (which kids might enjoy visit) and down to the sea shore at Ægisíða (Aegisida)(6). There you still might see some old fishing boats and the kids can run around on the shore. By looking over the ocean you can see Bessastadir, the home of our president, Olafur Ragnar Grimsson and his wife, Dorrit Mousajef. (7) End the tour in Vesturbæjar-sundlaug (west-city-swimmingpool) where it is nice to relax.