How to turn off flash on my Nikon D3200?

I can't for the life of me find out how to disable the flash, there's only one section in the menu that mentions flash but it is greyed out and won't let me enter it, any help would be greatly appreciated.

That is probably because you are using the green completely-auto mode. If you switch from the green completely-auto setting to the P for program setting then the flash will only pop up when you press the flash button.

Assuming it is like my old D60 then there are only 2 other differences between those two settings.

First the auto-ISO gets switched off. It is probably a good idea to learn how to use ISO manually anyway. To start off with keep it as low as possible unless your shutter speed is getting too slow. Start off with ISO 200 most of the time outdoors, switching to 400 on dull days or early light or evenings. Use ISO 100 when it is really bright outdoors. Go to ISO 800 when it gets darker in the evenings. Indoors you either want ISO 800 or flash with ISO 200. That's a bit of a generalisation, but it is a good start.

Secondly, with the P program setting, you can use the thumb wheel to change aperture and shutter speed settings. If you increase shutter speed it will automatically give you a wider aperture to keep the exposure right. If you decrease shutter speed you will get a smaller aperture. It won't let you go too far because it tries to keep the exposure right, so if you try to increase shutter speed but have already reached widest aperture, it won't let you increase it any further - unless you increase the ISO first.

Getting used to the "P" mode is a good first step towards learning what your camera is capable of. Learning to shift the shutter/aperture settings will allow you to achieve the effect you want. A faster shutter will help to freeze movement. A slow shutter will give blur if you want it, like those ghostly white waterfall pictures. A wide aperture will mean a smaller depth of field - so you can throw the background out of focus. A small aperture brings more of the image into acceptable focus.

Adjusting th ISO is the third part of the equation. A small ISO will give least noise in the picture, but higher ISO will allow you to use faster shutter speeds and/or smaller apertures. You will need that when there's less light.

To start with just switch to P and get used to adjusting the ISO setting. There may be an option of auto-ISO on the D3200 but I'm not sure about that.

You can't control the flash if your camera is in auto mode. You have to go to one of the Program, Shutter Priority, Aperture Priority, or Manual modes to control the flash.

Auto mode is essentially "Autopilot" and the camera controls all aspects of the exposure. Sometimes it makes the wrong decision, so you should attempt to learn how to use the advanced modes on your camera.

Pick up Bryan Peterson's book "Understanding Exposure" if you want to get out of auto and take your photography skills to the next level.