I'm no expert but here's my take on it. In no particular order...
They say that your shutter speed shouldn't drop below 1/focal length (x crop factor). So, of you have a 200mm lens, you shouldn't go below 1/200th of a second (or for a crop sensor half as much as that again, so 1/300s ish). But if you have VR then you can maybe drop that right down by up to 3 or 4 stops - 1/200 > 1/100 > 1/50 > 1/25th etc.
But, that may depend on how steady you are. And don't forget VR won't help freeze moving subjects! So, if you have a fast moving or jittery subject, you want a fast shutter speed anyhow!
Your aperture (F stop) determines your depth of field (DOF), as you know. So a wide aperture (say F2.8) will give a very shallow DOF while a small aperture (F22) will give a wide DOF. Some say not to go too small an aperture to avoid softening of the image due to diffraction. But, if it means you get the shot you want, then I'd say go for it!
Whether your depth of field or shutter speed is the most important depends on what you're shooting, obviously. If I'm trying to catch a bird (on 'film' of course, not literally) then shutter speed is the most important, so I'll personally then go into shutter priority and set a fast shutter. How fast 'fast' is depends on what it is and how much light there is. The faster the better for moving birds for example, even sitting on a branch they can be quite jittery! Aim for 1/500 or as much as you can before having to go to an ISO that'll make the image look poor anyway.
And talking of ISO. They say a grainy image is better than a blurry image and that's very true, but be aware that heavy cropping a high ISO image can be disappointing!
So, I normally walk around on aperture priority, at a fairly wide aperture - my 70-300mm is quite happy wide open - with auto ISO, if I'm just out for a walk with the family. I have my camera auto ISO set to 1/focal length for walk around with the kids stuff.
Perhaps one of my biggest lessons I had to learn was not to compromise shutter speed in trying to get really low ISO! Really, honestly - I was so obsessed with keeping my ISO low, that I had so many ruined blurry shots! Oh, and not to try and under expose to try and keep ISO down either - to keep noise down you want plenty of light on those pixels!
So, for me, to summarise:
* landscape: Aperture priority, F8/F10 (look up hyperfocal distance), let camera go down to 1/focal length shutter speed (unless you're on a tripod of course in which case you can go much slower) and low ISO
* wildlife, kids running about!: Shutter priority >1/500 or whatever you can manage! wide aperture (F2.8 or 5.6 or whatever), high ISO to keep shutter speed up
* portraits: wide aperture, slower shutter would be ok, low ISO
But rules are there to be broken and every situation is different!
I'll stand aside now, so someone who knows what they're talking about can come in...