So much of this depends on how heavy an object you're trying to move, and how fast...
At the very bottom end, an Arduino with a Motor Shield makes a decent minimum-cost starting point. Hobby servos are widely available in a broad range of sizes and power ratings (and prices), and their gearhead interfaces are standardized so it's easy to find CAD data for them if you need to 3D-print an adapter of some kind.
I second the SparkFun and AdaFruit recommendation -- they have all the components, and most of the component pages have tutorial videos and sample code to start from.
Okay thank you I'll definitely do that.