Goodness me, such heat generated over such a simple subject! I think you nailed it yourself when you asked about the trade off. In essence, it is up to you, it depends on what you are photographing and it depends on what "effect" you are trying to achieve.
I am not going to talk about when "I" change the ISO I shoot at because it isn't relevant. Instead, I want you to think about some scenarios.
If you are shooting some fast moving sports action and you want to freeze the action, you would use as wide an aperture as you can and as high an ISO as you can. If the light levels allow, you might want to reduce the ISO to improve noise while keeping the shutter speed high.
If you are shooting a waterfall and want to have a specific shutter speed, say one eighth of a second, in mind to give a certain "look" to the water, have a tripod and want good depth of field, then a low ISO is called for and will usually give you better dynamic range and less noise also.
Lets say you are shooting a landscape with some foreground detail you want to keep blurred and are shooting hand held. Common sense may dictate a certain f stop to keep depth of field narrow (for blurring the foreground) and a certain minimum shutter speed (depending on focal length and steadiness of your hands) to prevent motion blur. If that is so, then your only FLEXIBLE parameter is the ISO and you would choose the value to suit. Taking this hypothetical situation further, if light levels happened to fall during the shoot, you would compensate by increasing the ISO. If low noise, and good dynamic range and a good depth of field were paramount, then you would choose a low ISO, a high f stop number and a tripod!
In other words, you have three PARAMETERS for exposure to choose from - shutter speed, f stop and ISO. CHOOSE the important PARAMETER or PARAMETERS to give your photograph the "look" you WANT, and then select the other one or two parameters to give you the EXPOSURE you need. In short, in this example, choose the shutter speed and/or f stop you need for the shot and choose the ISO to suit, bearing in mind that the lower the ISO, the better the image quality.
I hope this helps.