If you go with Grbl, I strongly suggest you spend the extra couple of dollars and go with the mega version (running on an Arduino Mega 2560 - and while the official, made-in-Italy version is a nice board and I do like supporting the official Arduino project where possible; if you are short on funds you can get cheap clones on eBay, Amazon, Aliexpress, BangGood, etc; at the moment BangGood had some for $6.99USD).

The reason I suggest this is because the Uno version that runs on a ATMega328 has literally used up all the storage space on the board... there is no room whatsoever to add anything at all. However, the Arduino Mega 2560 has 8 times the flash storage space; leaving plenty of room for expansion (and more SRAM, too).

Also, a RAMPS board makes a nice platform for setting up a CNC; with plenty of IO for all the sensors and provisions for 5 stepper drivers (however, they are step-stick form factor drivers; and are only good up to about 2 amps per motor - if you need more amps, you will need external drivers, such as GeckoDrives or perhaps something TB6560 based).

The official version of Grbl for the Arduino Mega 2560 is here:
https://github.com/gnea/grbl-Mega/

Also, FWIW, I have my own version of Grbl-Mega; that adds support for 3D-printer derived user interface hardware (i.e, LCD & SD card support). Note that I do not recommend that you start with this, because it's still very much in development, but if you are curious to see what I'm up to with it, my version is here:
https://github.com/bdurbrow/grbl-Mega/

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Although steppers are not suitable for driving the spindle; some people do outfit their machines with servo motors for the main spindle drive. I was just looking at an ad for a 750w AC servo yesterday - for $599USD, if I remember correctly. There are several advantages of a servo motor: you can command a precise speed and as long the motor isn't overloaded, it will hold that speed exactly; while threading or tapping you can command precise synchronization between the spindle and the Z axis (so that you get exactly the thread you asked for; and so it doesn't cross-thread the part); and if you fit a brake to the spindle, the servo motor can move the spindle to a precise angle, clamp the brake; and you can use a drill head on the tool post to make a bolt hole pattern.

However, as noted initially - they are, for this scale of machine; quite an expense; and it is possible to get some of the benefit of a servo from a regular motor by fitting a sensor to the spindle, and having the CNC controller follow the spindle's rotation with it's Z axis movements. Note that the official version of Grbl does not have this feature yet; however there are other people who are experimenting with adding it.

One of my longer range plans is to make my own turning center; almost from scratch -- I would buy linear rails, motors, etc; but the frame of the machine would be completely DIY; and made out of steel and epoxy-granite. For this project, if it's not already in the official distribution of grbl by the time I get there; I will need to integrate someone's method of doing threading... probably based off of fschill's work. (https://github.com/fschill/grbl-Mega/tree/spindle_sync)

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A quick google search seems to show that the lathe you have is somewhat similar to a 7x12 or 7x14 bench top lathe such as those produced by Sieg in China (although from the photos it looks like the MD65 is better made?). There are many CNC conversions of Sieg lathes that have been documented on the internet; and although the exact details will vary; the general approach would still be applicable.

P.s. - lathes typically don't have a Y axis; Z is towards & away from the spindle; and X is in and out along the radius of part being turned. (Yeah I know - I thought it was a little odd the first time I saw it too)