Personally, I find aperture priority the least use for me. Sometimes, but not that often.
All methods can be manual. To me, as long as you know what the camera is doing and you're in control, it's manual.
Shutter priority, is just what it says. The priority is that the shutter remain at some shutter speed you wish, generally stopping action. If you don't want the shutter under 1000 of a second to stop a bird in flight, that's shutter priority, or the same goes for freezing kids or anything else where you don't want subject blur.
Aperture priority is for when the aperture setting is the most important thing to consider and freezing action is second choice or not important. That might be for portraits or where depth of field might come into play.
Program mode is for when you just let the camera choose. Freezing action or depth of field is not that important. You can shift it if you wish to empasize one of these things. You still have control.
I use program mode to tell me the exposure. I usually change to manual mode to lock in my choice based on what I found out. I then have control over freezing action or not and depth of field. This way, bright sunny backgrounds don't fool the camera. I also sometimes take a picture just to see the histogram. Once I do this, I don't need to change the exposure from frame to frame. Unless I move to different light, I leave that manual exposure alone, or adjust it a stop or so in some direction depending on need. I look at histograms fairly often to keep it honest.
If program mode tells me 125th of a second at f/8 and I want to stop action at 500th of a second. I know I can just move two stops. I'll change to 500th which is stopping down two, then open the aperture two to make up for it, or f/4.
You have to be careful and set ISO either manually or auto with a limit or your test exposure might be off.
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Cheers, Craig