Tomatoes are generally propagated from seed, although there are new hybrids which can't come from true seed. 

Propagation Methods:

By Seed:

Timing: When to start Tomatoes

• Start seeds indoors 6-8 weeks before last frost (2/18-3/4 in Redding, CA), although recommendations from seed vendors range from 4-6 to 5-7.
• Gel sowing takes 2-3 weeks less time (REOG)
• if growing in a heated greenhouse, start seeds in January (REOG)
• for unheated indoor sowing, sow 6-8 weeks before last frost (REOG)
• outdoor sowing is not generally recommended
• because tomatoes have a long growing season, it's helpful to start the seeds indoors so that when it's warm enough outside, the starts are ready. Starts from seeds sown indoors also tend to be healthier, stronger, and more resistant to diseases. (GV)

Pick a Growing Medium / Method:

• Growing Medium must be sterile.
• Types of growing mediums::
  • potting mix for eggplants & tomatoes (REOG):
    • 1-2 parts or 2-4 gals worm compost
    •    1 part  or    2 gals sharp sand or horicultural sand
    •    3 parts or    6 gals well-rotted leaf mold (or other peat alternative)
    • plus 2.5 oz ground limestone
  • Seed starting sponges: these are my favorite. just plunk them in a tray with cells, or better yet, a Styrofoam planting block then plunk your seed in the pre-drilled hole. they soak up moisture at just the right rate and have nutrients for the seedling. with the floating planting block, you don't have to water from above, which can damage the seedling. Also, there's less occurrence of stem rot. Also, no fungus gnat larvae to deal with. Park Seeds is where i get mine from.
  • Soil-less Growing Medium: usually consists of some combination of peat moss, bark, coir, perlite, and/or vermiculite. It's preferred over soil due to sterility and good drainage and better air space.
  • Gel Sowing: I've never done this, but it sounds like a fantastic way to do it, although you basically need a laboratory...sterile, great moisture & food control. Also takes up less space & quicker
  • Bubbling Water: this method is known as "Imbibing". seeds are enclosed in cheese cloth, then dangled in an aquarium or other water bath with air bubbled through the water. Very quick method, but you have to untangle the germinated seeds without hurting the roots, then pot them up. Haven't tried it myself.

Containers:

• flats filled with medium: seedlings should be potted up when first true leaves emerge.
• trays with cells or cell packs: depending on the size of the cell, they will need to be potted up once the first true leaves emerge
• 3 1/2" pots: no potting up necessary
• great to start in tub-shaped biodegradable pots that can be planted directly into the soil. Over time the pot rots away naturally.

Lighting

• Light: Tomatoes can be started in a sunny window sill, but i can never find a window sill big enough! 16 hours of light is required when starting seeds indoors. This is because indoor lights have less intensity than the sun.
• 6-12" above plants, and must be moved as the plants grow taller. If the lights are too high, then the plants get straggly.
• on for 12-16 hours per day
• Normal flourescents work until the plants are older, then they need additional wavelengths of light. I am working on an article that will go into lighting requirements in depth, but this is all you need to know for starting the seeds.
• When seedlings appear, move the trays or modules somewhere with bright light and a constant temperature of 70-81 degrees F.

Temperature:

• Air temp should be at least 60 degrees, but i think they like it better in the mid-70s. over 80 is too hot.
• Soil temperature needs to be 70-90 degrees for all nightshades.
• sowing is best done undercover, either indoors, in a green house, or inside a cold frame. 
• gentle heat from under is best. heating mats are sold just for seed starting.
• Tomato seeds are unlikely to germinate at temperatures lower than 59 degrees F, so you may want to put them in a heated propagator.

Water

• water very carefully: a light sprinkle so you don't float the seeds.
• water from the bottom, if possible

Humidity & Air Circulation:

• humidity should be high, but the higher the humidity, the better the air circulation needs to be. I use domes to cover the seed trays to keep it humid, with the vents open, to get some circulation. As soon as some of the leaves are touching the dome, i remove it. If I don't remove it at this point, the leaves touching the dome rot.

Seed Depth

• If using the seed starting sponges, just drop the seed down the hole; don't worry that it's going too deep.
• Plant tomato seeds 1/4" deep (G101)

Germination

• Days till germination:
  • 7-14 days (G101)
  • 6-10 days (TKG)
  • if imbibing: 3 days (TKG)
• Germination temperature: 
  • 70-80 degree F soil (G101)
  • at least 60 degrees F (REOG)
  • 80 degrees (TKG)

Progagating By Division

not possible because of root system

Propagation by Cuttings

• side shoots are easily rooted by cutting from plant when 6" long.
• insert into 3" pots filled with eqyal parts leaf mold & sharp sand.
• cover with plastic bag or bottle (bell).
• provide bottom heat at 60-62 degrees F early in season
• cuttings root in 10-14 days
• pot up or plant out. 
• remove the firs truss of flowers if the plant isn't growing strongly

Seedlings:

• Seedlings can withstand cooler soil temperatures (REOG)
• water tomato seedlings sparingly; kept on the dry side, roots will develop but plant will stay short. this gets the plant off to a better start when planted out.
• Transplant after about 8-weeks in late spring or early summer, depending on weather. (GV)

Transplanting or Potting Up

• Transplant 1-2 times to avoid root crowding, before planting in the ground (G101)
• Set outdoors 1-week after last frost date (4/22 here in Redding, CA) (G101)
• Should be planted outside before they flower. pinch off flowers if they bloom before getting planted
• Harden off:
  • when daytime temperature is above 50 degrees F
  • take outside for the day, then bring them back in at night
• When the seedlings have two or three true leaves and are large enough to handle, carefully prick them out into individual 2-3" pots or biodegradable tubes. Keep them in a warm, bright, well-ventilated place.

Seed Saving:

• Tomato seeds are easy to save, keep pure & harvest (REOG)
  • crush tomatoes
  • let ferment to kill some of the diseases that could be present
  • soaking seeds for 25 minutes at 122 degrees F eliminates target spot & bacterial canker
  • wash & dry well
• Very high success rate; collect seeds only when the fruit is very ripe. Cut it in half, pry out the seeds, leave to dry on paper towels. Don't attempt to save seeds from F1 hybrids; they are unlikely to inherit the same qualities of the plant you harvested them from. Even non-hybrids may cross-pollinate. If you want to be sure to get non-cross-pollinated seeds, grow the plants well away from other cultivars. (GV)

Sources: