You might know a lot of this is you have been reading up on it.
Mach 3 puts out signals on the LPT1 - the old computer printer port. There are only 12 output lines and 5 input lines ( It is the input lines you might run short of) - so run through what you might need.
Each axis requires 2 wires (plus a return wire) to drive it, one wire drives the step, one wire drives the direction. If you have four axis you need eight wires etc. Don't just allow for three axis - most people eventually want a rotary table and other goodies.
Outputs must also drive things like the spindle control (normally three wires) coolant (one or two wires) axis enable wires (if required) one for each axis. etc.
Inputs are for limit switches, home switches, spindle speed, and any other switch you might want to put on your lathe like an "emergency stop" button or other safety device.
You can see that it is very easy to run out of wires, when your system gets a bit advanced - and you end up having to add other ports to your computer.
Many firms offer driver boards and breakout boards etc. A breakout board merely provides an easy way ( usually screw terminals) of connecting your bits and pieces to your computer. Many breakout boards have electronics on board to isolate signals and provide a 5 volt power supply for switches etc.
The disadvantage with a lot of breakout boards is that they are pre- wired and fixed in their applications - rather like painting by numbers - fine to get started with, but in the end, very limiting.
The Mach 3 admin boys are going ******** at the moment about a new interface board called a Smooth Stepper. This is a board that provides inputs and outputs (more than you need), it provides on board pulse engines to drive your axis, which means that your computer does not have to generate the pulses and can get on with calculating the job. I understand that it is opto-isolated , etc etc . The beauty about it is that it connects to your computer via a USB lead. I would recommend that you look at one of these ( if you are starting from scratch).
Whichever type of board you start with - you then need to interface this with your machine.
Geckos are good drives and I am pleased with mine - so I cannot crticise your choice. Use seperate drives - Do not get a single board that has breakout board and three drives all in one - if one goes faulty it all goes faulty. If you have seperate units you can change anything at will.
The output from your breakout/smooth stepper board will drive the Ghecko board directly. Mine require a 5 volt connection, a "step" connection and a "dir" connection to each board. The inputs to the Gecko are isolated from the rest of the board to protect the low voltage side of the system.
I have looked at the Grainger website and your choice of axis motor. I am at a bit of a loss. These appear to be a straight DC brushed motor, and there is no mention in the blurb of feedback for positioning, There is mention of a tacho - is this for positioning.
Mach 3 gives out step and direction signals. It is preprogrammed with the number of steps per inch of axis movement, so if you want to move one inch - it puts out say 60,000 pulses and says - "there you are - one inch". With steppers you hope the motor got all the pulses, but with servos (as I understand it) there is a measurement system either inside the motor, or external to, which checks that your motor has gone the relevant distance. I do not see that system mentioned in the Grainger.
If you read the PDF file on the Gecko 320 it tells you how to wire them, and also about the feedback circuits. You can get external feedback circuits, but I thought that servo motors ( marketed as such) came with them ready installed integral to the motor. I would check the motors.
If you need any more - come back again.