First year I tried this, I killed approximately forty plants. This is the no-mistakes version.
Tomato seeds want warmth and moisture to germinate. Once they sprout, they want light and not too much water. Most failures happen because people get one of these wrong.
Count back six to eight weeks from your last frost date. For me in zone 7, that's mid-February. For colder zones, push later.
Cell trays with seed-starting mix (not regular potting soil — too heavy). Two seeds per cell, planted ¼ inch deep. Water gently from below if possible. Cover with the clear lid that comes with the tray, or plastic wrap.
Remove the cover. Move to the brightest light you can find. A south window helps but isn't enough for most homes — a cheap LED grow light makes a huge difference. Keep the lights an inch or two above the plants and raise them as the plants grow.
Water from below. Bottom watering means the soil stays moist but the leaves stay dry, which prevents fungal issues.
Two weeks before transplant, start putting them outside for an hour, then two, then a full day. Skipping this step is how you cook a six-week-old plant in an afternoon.