Growing tomatoes from seed takes time and care, but ultimately it is not hard and the results are well worth it.
You will need either a grow light setup (cheap shop lights) or a south-facing window that gets plenty of sun (if you live in the southern hemisphere, you will need a north-facing window).
Use this instructable as a guide to growing your own tomatoes from seed.
UPDATE: Check out my new Web site on growing tomatoes. Grow Your Tomatoes
Step 1: Get the seeds
Research the seed company you are buying from to make sure they are reputable and that you will get good seeds from them.
I'm an heirloom grower, but you can choose to grow either heirlooms, hybrids, or some of each.
Step 2: When to plant the seeds
To find your average last frost date, go to http://cdo.ncdc.noaa.gov/cgi-bin/climatenormals/climatenormals.pl?directive=prod_select2&prodtype=CLIM2001&subrnum%20to%20Freeze/Frost%20Data%20from%20the%20U.S.%20Climate%20Normals
I live in Middle Tennessee and start my seeds in mid- to late-February.
Step 3: Start the seeds
I use egg cartons to start my seeds in. You can use either the clear plastic or Styrofoam cartons; do NOT use the paper ones. Fill the trays with seed mix and firm the mix down into the cells.
If you are growing multiple varieties, you will need a labeling system to keep track of what tray contains what variety. Use tape, plant tags, etc to mark the trays. Be creative - do whatever works best for you to keep track of the varieties.
Plant the seeds about 1/4 inch deep, 2 seeds per cell. I use a pencil with the tip broken off to make a 1/4" deep hole in the center of each cell, and I drop 2 seeds into each hole and firm the mix around the seeds to completely cover them.
Step 4: Germination
In anywhere from 3 to 15 days, you should start to see tiny seedlings emerge.
When your seedlings are up and the first 2 leaves (cotyledon leaves) start to open, you will need to put your seedlings under a light.
Use a cheap fluorescent shop light for your seedlings. I use 4' fixtures that take 2 bulbs each. You can use regular fluorescent tubes, or ones specially made for plants. I use GE "Plant & Aquarium" tubes in my fixtures.
It's very important to keep your seedlings within 4" of the lights, preferably closer. If you keep the light too far from the seedlings, they will get very "leggy" - tall and skinny - and might collapse.
Keep the seed starting mix moist but not soggy, and water whenever the surface becomes dry to the touch (but NOT completely dry).
Step 5: Care for the young seedlings
Make sure they are kept within 4" of the grow light(s). You can adjust the chain that the light hangs from, or you can put the seedling trays on books or boxes to adjust their distance from the light.
Make sure your grow light setup is in a room where it won't get too hot (80+ deg F) or too cold (below 50 deg F).
You will want the seedlings to be easily accessible, because you will be watering them often (every couple days).
Monitor your seedlings and make sure they are growing well. The cotyledon leaves should grow up to 1 inch wide each, and should be a healthy green color.
Damping off can be a problem - this is a disease that causes young plants to collapse at the soil line and die. If any of your seedlings damp off, remove the infected plants and the mix they grew in to prevent spreading the disease to other plants.
Step 6: Potting up
I use 16-ounce disposable plastic cups. These work well and are cheap.
Fill the cups with Miracle Gro Potting Mix or similar potting mix. I use Miracle Gro because it eliminates the need to apply fertilizer manually.
Use a pencil to make a hole in the center of the mix in each cup. The hole should be about 1 inch wide and 3-4 inches deep.
Choose the best seedlings to pot up, and discard the rest. If you pot up more than you can grow in your garden, just give away the extra plants when they are bigger.
Carefully loosen the seed starting mix around your chosen seedlings. Gently scoop out each seedling, being careful not to damage the roots or stem. Tap off excess mix from the roots so they will fit easily into the hole you made in the mix in the larger cups. Do this one at a time, and when a seedling has been uprooted, put it in the larger cup immediately.
Firm the potting mix around the roots and the stem of each seedling. Bury the stem all the way up to the cotyledon leaves - roots will grow from the stem and benefit the plant.
Label your cups with the variety name of each plant. I write on the side of the cup with a Sharpie.
Thoroughly water all the cups. Make sure you don't splash potting mix all over the seedlings when you water them.
When the seedlings are all potted up, put them under the grow light(s) and keep them within 4-5 inches of the lights.
Step 7: Planting out
Dig a trench about 1 foot long, with one end deeper than the other. Carefully remove the plant from its pot and loosen the root ball. Place the root ball in the deeper end of the trench, and lay the seedling on its side with the stem in the trough.
Remove all the leaves from the part of the stem that is in the trench, and leave the top few leaves on where they will be above the ground. Bury the roots and bare stem in the trench, leaving only the top few leaves sticking out. Don't worry about their being sideways - the plant will correct itself and grow upwards within a few days.
Trenching plants allows roots to develop along the entire buried portion of the stem. They say this increases yields - I haven't done any tests of my own but it makes sense. More roots allow for more nutrients to be absorbed.
Repeat the trenching process to all your plants, keeping them spaced at least 2 feet apart (if you plan on pruning) or 3-4 feet apart (if you won't be pruning).
When all the plants are trenched, water them thoroughly and add mulch if desired. Drive stakes or cages into the ground, making sure you don't puncture the buried stem.
Step 8: Care in the garden
Keep the plants watered, but not overwatered. Don't let them dry out or inconsistently water them. Not enough water can cause fruit problems like Blossom End Rot, and overwatering can cause the fruit to crack.
To prune your plants, pinch off the suckers (shoots that come out from between each leaf and the main stem). You can let a few suckers grow for more fruit per plant, but as a general rule, the more fruit you allow, the smaller they will be. Last year I let one sucker grow on each plant for a total of 2 stems per plant. The rest of the suckers were pinched off as they grew.
If you prune your plants, tie the stems to a stake. If you don't prune them, you can let them sprawl on the ground or place tomato cages around the plants.
Step 9: Harvest
You can cut or twist the fruit off when it is fully colored. Some heirloom varieties ripen green or have green shoulders when ripe; Google the varieties you are growing to see what they look like when ripe.
From this point forward, it's mostly watering and picking fruit till the season ends.
Here are a couple useful links to help with any problems the tomatoes may have:
Diagnose tomato disorders/diseases