The Tampere City Museum (Tampereen kaupunginmuseo) within Vapriikki covers the urban history of Tampere from its founding in 1779 through industrialisation and into the 20th century. It is the most straightforward way to understand the city's development — the factory period, the Civil War, and the post-war reconstruction are all covered with good use of photographs, personal objects, and maps.
A good first stop within Vapriikki if you want context before exploring the city. Included in the combined ticket.