We order our seeds from Seed Savers which is an incredible organization specializing in heirloom and organic seeds. The variety is fantastic, and choosing amongst the colorful catalog is both a difficult and delightful task. We not only order vegetable seeds from them, but also herbs, exotic annuals, sunflowers, and for the first time this year, fruit plant seeds.
After the seeds arrive, we convert our sunroom into our mini-growing lab...which from the street looks sort of suspicious and I often wonder what passersby think. I got this handy newspaper potmaker for the plants and while watching TV I sit and make pot after pot, writing on the side what seeds I will grow in it later. There is special seed starting soil that can be bought at Home Depot or the like which works great. Once the pots are made and filled with soil, 2-3 seeds are sprinkled in each one, lightly covered with soil, and placed on the radiators (or another warm, sunny location) until they start to germinate. Before we got the newspaper pot maker, we used a peat pot kit, like this one, which also works great.
At this point, we set up a table with a long rectangular flourescent light that suspends from the ceiling over it. The light should be quite close to the seedlings, just a few inches above them, otherwise they get "leggy" trying to reach for the light. Once they grow, we raise the light a bit.
I keep them lightly watered and try not to check on them every second of every day.