The first thing to mention about National Museum of Scotland is, that it is free. I was at the museum in 2017 and the section that spoke to me the most, was the Science and Technology Gallery. Here was planes, motors, electronics, race car simulators, robots and a bunch of things you could interact with. Also the Natural World gallery is worth my recommendation. The many skeletons and life-like figures are fantastic Of course Scottish history and archaeology is on of the topics in the museum. There is also a vast number of other exotic cultures represented in the World Cultures Gallery.
Calton Hill in Edinburgh is mostly known for its great view over Edinburgh. You will not find a guidebook without a photo similar to the one above. Besides the Dugald Stewart monument (a memorial to the Scottish philosopher by the same name), you will see many other landmarks on the hill and below. One of them is the National Monument which was never completed, as only half the money to build it was ever collected.