2a. The Golden Circle - Thingvellir

In the year 930, the 'Alþingi', one of the oldest parliamentary institutions of the world, was founded at Þingvellir. It is a beautiful place with a big rift 'Almannagjá' caused by the movements of the Eurasian and North American continental tectonic plates. It is now on UNESCO Cultural and Natural Heritage list.
Following link will show you a map of interesting places, hiking trails etc.
You can either stop at the Visitors center on top of Almannagjá and walk down the clift to take a look at all the places, or drive on a bit further to Hótel Valhöll where you are closer to the lake.
Walk along Almannagjá to the river 'Öxará' with the waterfall 'Öxarárfoss'. In the river you see 'Drekkingarhylur' ('drowning pool'), where female lawbreakers were drowned, especially if people believed they were withces. Take a look at the assembly site, including the Law Rock (Lögberg) and the Law Council, where the Alþing performed its duties. Closer to the lake you can see the church. Close by there is a rift filled with water, called 'Nikulásargjá' or 'Peningagjá' ('Penny Canyon'), as it is littered with coins at its bottom. Legend says that if you throw a coin in the water and make a wish, it will come true if you can see it hit the bottom.
There are a lot of nice walking trails at Thingvellir, marked paths in the lava leading to old ruins where people lived many years ago. Those walks are not very far, maybe 1-2 hour in the whole.
It is also very nice going down to the lake. Many people go there fishing and if you have the right equipment you can pay for fishing for one day. It is also quite all right to wade out in the lake, throw stones and play around. Down by the lake is ideal place for picknic.
Is is also possible to go horsebackriding here.