When I was learning to move out of auto modes and to this day I still use Shutter Priority and Aperture Priority. I would do loads of tests and check them out on the PC rather than the LCD, as I found that worked better for me.
For example when checking out the different shutter speeds, I would either take shots of the kids or wife, running or walking around both inside and outside making a note of which settings caused less blur but remained non grainy.
Similarly when playing around with the Aperture I would take a number of shots without moving the camera and see how aperture cam into affect from a DOF point of view.
I guess my best advice is to practice as much as poss and see the effects of your shots. Also I think it's great to check out other peoples photo's to see what you like and analyse the photo i.e how they framed the shot, the lighting used, angle of shot etc