My 4 X 4 lighting fixture in the kitchen has flickering fluorescent bulbs. Is the problem the bulbs themselves or the ballast? What wattage bulbs should I use? 32 or 40 watts?

Flickering lamps can be caused by defective lamps or ballast.

If the lamps have any age to them, cheapest method is to install new lamps. Lamps must match the ballast, so, if there are 40 watt T12 in the fixture now, the ballast should be for 40 W T12 lamps.

32 watt lamps are a newer generation of lamps, and must use 32 watt T8 ballast.