Ok, well there is no right or wrong way to do this of course, you just need to find what works best for you. For me I've found that center weighted average is the best overall metering mode for the kind of pictures I shoot. Matrix metering tries to average the light of the entire scene which doesn't always produce the best results when your shooting into a shady area. Spot I've found can lead to some unpredictable results depending on which spot is being metered, so for me at least center weighted average produces the best overall results in most of the situations I find myself shooting in.
For critter/action shots a lot of folks really like A - Aperture priority mode. Me I prefer S - Shutter priority mode. I have the camera setup by default to choose the ISO automatically, I set my shutter speed and the camera selects the aperture and ISO for correct exposure. I can see both the aperture and ISO settings in the viewfinder, if I think my aperture setting is a bit to high and I want to open the lens up a bit for more background separation I can just dial up a faster shutter speed until I get my aperture where I want it to be and I'm ready to shoot. If I think my aperture number is too low and I want to increase it I can increase my ISO or lower my shutter speed, or a combination of both until I get what I'm looking for as far as aperture.
Since my D5200 has only one command dial I find this the quickest and easiest way to have control over my settings without having to access any menus or mess with anything other than the command dial and the function button if needed.