pH, Ammonia and Nitrate are the main ones really, I don't use any of my other tests but then again, I hardly use the main ones either.
Whilst cycling, only change the water when the concentration of Ammonia, Nitrite and/or Nitrate gets too high for the fish to handle. If using the fishless cycling method, no water changes are needed until the end.
I stopped using a gravel vac as soon as I got serious about plants. If you feel the need then only siphon the muck out of the gravel away from the base of plants or else you are taking the free nutrients (from fish waste) away from them and disturbing their roots. As Dennis said, you don't need to be too meticulous about gravel vaccuming.
HTH