You might want to spend some time browsing the exhibits at Ural Geology Museum, Yekaterinburg History Museum, or Boris Yeltsin Presidential Center. While you're exploring, swing by Church on Blood in Honor of All Saints Resplendent in the Russian Land, Rastorguyev-Kharitonov Palace, and Novo-Tikhvinskiy Convent to learn a bit about the area. You can watch performances at Yekaterinburg Circus, Puppet Theatre, and Yekaterinburg Theater for Young Spectators if you're interested in local theater.