I have a Sigma 56mm f/1.4 and Sony a6400. I'm looking into buying a diopter because it seems to be relatively cheap and can reduce the minimum focus distance while also zooming in, which I both want. Which diopter specs should I look for? I want <25cm focus distance if that's realistic.
Because of the very simple optical design of the diopter (or, technically more correct: close-up lens), the image quality will suffer a lot, e.g. from heavy chromatic aberration. The results will be like using the cheapest, crappiest lens instead of your quality Sigma one.
Instead, I'd recommend an extension tube. IMHO, they are a much better value for the price: better image quality and much more flexibility.
An extension tube will not interfere with the optics of your lens, just use it a bit outside of its designed distance range. From my experience with other lenses, I'd expect quite decent results from your 56mm prime.
A 10mm extension tube, used with your 56mm prime, will give you a focus distance range of roughly 20 to 30cm. And they often come in pairs or sets, so you can achieve various extension values, meaning even closer distances possible.
And they only have to match your body mount, and can thus be combined with whatever lens matches your body. The close-up lens, on the other hand, can only be used on lenses with the same filter diameter.
When shopping for extension tubes, just make sure that they are "automatic" ones, connecting the camera signals to the lens, and that they have a decent mechanical quality (e.g. looking at user ratings in online stores). My set for a Canon EOS-M was in the $20 price range, and I'm completely satisfied with them.
You can mount a close-up lens (diopter) on a lens of any focal length. These supplemental close-up lenses work exactly like reading glass spectacles that allow older people to read with the book or newspaper close at hand. In fact, you can go to the drugstore and handhold one lens of a reading glass before your camera. This will allow you to preview the effect. The power engraved on the reading glass is the diopter measure of the lens, same as the power of the one you buy for your camera.
The diopter is a unit of power (focal length), is used by eye doctors when they write prescriptions for corrective lenses. 1 diopter = 1000mm, 2 diopters = 500mm, 3 diopters = 333mm, 4 diopters = 250mm, 5 diopters=200mm.
If you mount a +3 close-up (333mm supplemental before your lens, you grant the ability to focus closer to your subject. With a +3 mounted and the camera set to infinity focus (∞), objects will be in focus 333mm forward of the lens. Using the camera’s focus knob, you can focus even closer. Your revised focus range is likely 1 diopter = 333mm to 250mm.
Close-up supplemental lenses can be a cheap single element or a good quality two element achromatic (without color error) lens.
Mounting a drugstore reading lens is an inexpensive way to preview what a close-up does. After you decide the power you need, buy a good two element, I suggest a +3 for your first purchase.