Servos are a good suggestion, and MichaelMeissner gave some excellent advice and resources on how to use them well in your desired applications.
Yet where feasible relays and solenoids would, IMHO, add to the steampunk ambience. This is because versions of these electromechanical components were actually around during most or all of the Victorian Era. Indeed since a significant part of steampunk deals with achieving, or at least appearing to achieve, modern functionality with 19th century technology; I would suggest researching how these parts were used, focusing on what modern electronics replaced them and why.
Occasionally they could even work better for a specific effect, for example for an automated hat tipping device you could use either a small servo or a small solenoid. Although a the solenoid would just need an extension to attach it to the hat, whereas a servo would likely need a more complex mechanical linkage between the two due to motion being rotational rather than linear.
However, practicality and budget concerns should trump vesimultude when they are in conflict. 