Choosing the right stepper

http://www.geckodrive.com/support.html/
section 9 should give the answers you are looking for.
Not a fan of combo machines and the price of cnc controllers have come down you can buy a whole cnc kit with controller board, end stops and motors for like $ 50 you can buy a small router pre build for under 200 and a small 3d printer for a similar price so make sure you know your options before you try to reinvent the wheel or in this case 3d printer / cnc machine.
You can build a GRBL based cnc  machine with a UNO boards and daughter board and a 3d printer from a an arduino mega and a ramps.

Interesting design concept
reminds me of this one originally designed to have a cnc capability. But the manufacturer abandoned the cnc head in favor of a router kit.
This kit is only $ 499

More ifo build manual  etc at this link
https://drive.google.com/drive/folders/ … sp=sharing

I considered a scratch build at one point but decided the time  it would take to work out a design then source all the parts was more than i wanted to get into.Decided a kit build was the way to go.
This is your journey will help in whatever way you need.

Any torque capability in excess of what the application requires comes at the high cost of
unnecessary motor heating. Excess torque capability beyond a reasonable safety margin will never be
used but will exact the penalty of an oversized power supply, drive stress and motor temperature.

The place to start is to determine the load torque in oz/in, including the torque necessary to
accelerate the load. The next step is to come up with the maximum speed the application has to operate
at in full steps per second using the formula below. RPI is the revolutions per inch after the motor turns
through the transmission, RPS is revolutions per second and PPS is the number of pulses per second
from your step pulse source.
(Desired IPM * RPI) / 60 = RPS
RPS * 200 = PPS
Multiply the PPS value by the number of oz/in determined previously and divide the total by
4506. The answer will be how many watts mechanical are required from the motor to meet the load
from the application. When picking a motor, choose one with 40% more than the calculated power.
Below is an example of the equation completed for a load requiring 450 oz/in with a 3 TPI leadscrew and
a desired IPM of 300.
(300 * 3) / 60 = 15
15 * 200 = 3000
(3000 * 450) / 4506 = 299 oz/in
299 * 1.4 = 419 oz/in
As you can see, you will want to use a motor with a rating of 419 oz/in for this application.

You need to know how much torque is required to lift the or lower the bed mechanism or  gantry and how fast you want/ need to move it.

also nema 23s tend to need like 3 amps of power way more than pololu shield will provide .
also keep in mind that the nema size has little to do with torque of a motor you need to make sure you motor has enough torque  but does not overdraw amps and blow a driver.
Also having 4 lead screws vs two should not make that much difference in motor requirements.

You may need to build you design measure required torque then select the motors .
I can not tell you what motor you need with no data to work with.

Based on experience and observations of machines a nema 17 should be plenty for what you propose but again you need to take some measurements and  do some math.
If this is your first 3d printer I suggest just buying one or at least a kit  rather that doing a scratch design and build.

tin

Hello Tin, thanks again for your expert explanation on this topic. I must say that I don't have any idea about what will eventually be the amount of torque needed to turn those four screws but the weight of the bed mechanism including all X&Y motors on it could be around 7kg max or 15 pounds. The screws would be one of these: https://tinyurl.com/ycs6ha77 and as for the speed it seems to me that the Z axis is not so important as the other two. I'm aware of the Nema standard but not so much experienced in interpretation of their values. Using knowledge from posts like yours I would conclude that this nema 17 with higher torque would be the equivalent of some nema 23 with similar values but when it comes to heat dissipation would it not be better to have a more massive motor? The one that can take higher load even it is a overkill for the task. If it is not so important then I stick to the smaller one because I'm unfortunately on a tight budget. And as for the drivers..hmm I'll stick with the Arduino but addressing your experience, do you have some recommendations?
Regards
Darius

Here is another option you might want to check out. They sell a CNC Kit and a 3D Printer kit.

I was looking into the 3D Printer kit for while for my own use.

http://www.pibot.com/cnc-store/cnc-electronic

Sorry about the second link  that kit is not really suitable for 3d printers . I inserted a link including a ramps card and a 2560 board .And even the new link only has 3 motors  for a 3d printer you need 4 motors for x,y,z and e. you also need input and output for at least 1 extruder and the heat bed.
also here is another link that may help.With motor selection.
http://reprap.org/wiki/NEMA_17_Stepper_motor