Yekaterinburg began in 1723 as an ironworks with a dam, and it is still a factory town that grew a city around itself. Peter the Great's wife gave it her name, the Soviet Union renamed it Sverdlovsk from 1924 to 1991, and it sits on the Iset River between European Russia and Siberia with about 1.5 million people. It is also where the last tsar and his family were shot in July 1918, which shapes how many Russians see the city.
It isn't a postcard city, and it knows it. The center mixes 18th-century factory history, a surprising block of 1930s constructivist architecture and glass towers from the 2000s. You come for the stories: the dam that started the city, the church on the site of the Ipatiev House, a president's museum, the leaning Demidov tower in Nevyansk, the stones the Urals are famous for. Three days is the right length. The center is compact, the museums take half a day each at most, and the one day trip fits in a day. A fourth day only makes sense for Olenyi Ruchyi, or as the start of a trip on to Kazan.
Day 1 is the old center, the Red Line and the evening skyline. Day 2 is the Europe-Asia obelisk, the Church on the Blood, the Yeltsin Center and the Geology Museum. Day 3 goes north to the White Tower, Ganina Yama and Nevyansk. Start on a Tuesday or Wednesday: day 2 should fall on a Wednesday to Friday, because the Geology Museum is reported open Wednesday to Sunday only. Read the tips first for visa, cards and internet, and see Russia for the country. For a longer trip, 5 days in Moscow and 5 days in Kazan are the neighbors on the same railway.