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Deep rooted plants such as tomatoes, zucchini, corn, and perennial fruits and vegetables, do best in planters at least 22 inches tall, or shorter if the ground below is usable.
Starting seeds is always a delicate part of growing anything, and learning how to grow tomatoes is no different. Start with a regular nursery tray full of your soil mix. Tomato seeds should be planted 1/4 inch deep, and about 8 seeds/inch. You may want to cover them for the first few days to keep them from drying out. If your seed starting tray does not have a lid, a sheet of plastic wrap will do fine.