When starting seedlings indoors you really need some type of artificial light. If your seedlings don't get sufficient light they can get "leggy" and weak. This is the grow light system that I personally own: Hydrofarm Jumpstart Grow Light.
This year I started my seedlings in Jiffy peat pellets, but really you can start your seedlings in many things from seed trays like this or even plastic cups.
Once the seedlings have started to sprout you want to take the cover off. Too much humidity can cause fungus and bacteria. At this point you want to make sure you keep the light about 2 - 4 inches away from the plants. Because plants indoors aren't getting natural sun, you will want to make sure you plants are getting about 16 hours of light. A lot of people put their lights on timers so they don't have to remember to turn them on and off all the time.
Today my seedlings are at 3 weeks and 2 days of growth. When they were about 3 inches tall I transplanted them to some plastic cups that I got for cheap. Unfortunately this all I have room for with this light system. Next year's project will definitely be getting more lights to do peppers and start my cucumbers indoors.
I hope to get these in the ground sometime after April 1st.