In short, manual vs automatic ISO setting has nothing to do with noise.
I'll try to give a short summary about how noise works, which you help you understand how much noise you will get in different situations.
How is ISO related to noise?
The ISO value that was used is a good indicator of how noisy the image is assuming that the photo was correctly exposed. This is why you hear that "high ISO = high noise" and "low ISO = low noise". But it is important to keep in mind that this applies only when the photo was correctly exposed (i.e. not too dark, not too bright).
Where does the noise come from?
From the above, you might conclude that you can control noise by setting a specific ISO. But stating it this way is misleading at best. Noise is not a consequence of the ISO setting. It is not even coming from the camera. Noise is inherently present in the light.* The relationship is as simple as this: "more light = less noise", "less light = more noise".
You control the amount of light reaching the sensor through the exposure time ("shutter speed") and aperture. For any given amount of light that was collected by the sensor, there is an ISO setting that will result in correct exposure (i.e. desired image brightness). If there was too little light (which also means high noise), you can compensate by setting a higher ISO. Therefore, both the amount of noise in the image, and the correct ISO value for proper exposure, are a consequence of how much light reached the image sensor.
In short: if you want less noise, find a way to let more light reach the sensor (bigger aperture, longer exposure time, or use a light source).
What happens if you set the ISO manually?
If you set the ISO to a low value manually, and let the camera determine the exposure time and aperture automatically, then one of two things may happen:
- The camera compensates by increasing exposure time and aperture. Depending on the available light, you may be risking getting a blurred image due to camera shake. However, you will indeed have less noise.
- The camera refuses to increase exposure time beyond a limit, and you get an underexposed photo. There is not enough light, so there is no way to avoid noise.
* This type of noise is called shot noise and it is the dominant noise source when working with modern cameras. Yes, there are other noise sources too, related to the workings of the camera, but as a first approximation you can ignore them.