Granny Shifting is up-shifting very slowly and carefully while pushing the clutch all the way to the floor for each shift.
A "racer" will only depress the clutch (if (s)he uses it at all) just past the "friction-point" while upshifting. A really good driver does this so smoothly, you hear the shift, but don't feel it, unlike the "herky-jerkys" of a bad shift.
Double-Clutching , (usually used when downshifting, not upshifting) is depressing the clutch, shifting into neutral, stepping on clutch again while matching revs, then shifting into the next lower gear (happens much quicker than it sounds).