Tomatoes need warm 75 to 85 degree conditions to start germinating. Put the containers in a warm place where they'll get bottom heat like on top of the water heater or refrigerator or use a fluorescent shop light suspended just 1 or 2 inches above the container and it will provide warmth . Keep the container moist, but not soggy. You can cover it with plastic wrap or an old piece of rigid clear plastic to conserve moisture if you like, but be sure to pull it up to check daily to be sure they aren't drying out. Water as necessary with a very gentle spray of water. If container should get too dry, you'll need to set it in a pan of water so it can soak up water again from below. Expect germination to take 5 to 10 days. Don't keep your containers in the windowsill during the germination period; cold air at night will affect germination. Check often!
Just as soon as any baby seedlings begin to emerge above the soil level, it's critical to give them light right away. Remove any covering immediately and provide a strong light source. While a south-facing windowsill is traditional, it's far from ideal, and dimly-lit plants become tall and spindly. I like to start my containers from the beginning under grow lights or a simple fluorescent shop light suspended from chains so I can move the lights up as the plants grow. The fluorescent lights under your kitchen counter will work very well for this. Raise your flats closer to them (4" to 5") with some bricks or fat cookbooks. Tomato seedlings grow best in the 65-75 degree temperature range.