Controlling the speed of an AC induction motor can only be done with a VFD (Variable Frequency Drive). You have to take the AC frequency (60Hz in the US), rectify it, then spit it out at a different frequency. Not at all a trivial problem so you won't find any simple circuits/kits/etc. to do that.
The good news is that VFDs typically include some type of control interface (usually multiple) and the Arduino would be fine for that problem. But of course nobody here can tell you how to interface with a VFD you haven't chosen yet.
Here's an example of a 3 phase VFD. It features either an RS-485 interface (requires just a TTL to RS485 chip, very cheap) or four inputs that act in a binary format to choose one of 16 speeds. You'd need a basic 4-relay board ($5) to use that interface.