First You can find a lot of info that answers a lot of questions you may have from reading the FAQ's at the top of this and other forums.

Most here who grow under lights use shop lights with 2- 40 watt flourescent bulbs. The fixtures should be suspended by chains or fixed so you can raise and lower them. I have a greenhouse and don't grow inside but from what I've read here the light should be about 1/2 inch and certainly no more than 2 inches above the plants. If your plants start to get leggy they are not getting enough light.

What else you need--trays, a humidity lid, pots, a good fast draining soiless potting mix--please stay away from miracle grow products--I hate them.

I use pro mix BX. It can be expensive in small bags but not so much by the bale. It will keep for years if it is kept very dry. I store my excess in a garbage can with a tight fitting lid.

The humidity lid is used to keep seeds moist until they germinate and then is removed so you don't get damp off. That is a mold that kills seeds and seedlings. Damp off is also why you don't overwater. Before you plant mix the soil and water so it's wet right through. Squeeze a handful and a little water comes out not a stream. That is perfect moisture for planting seeds. Try an experiment--put the dry mix in one pot and the moistened one in another. Now lift them--see how much heavier the wet one is. That's your test for watering. A pot may look dry on top but still have plenty of moisture underneath.

Pots--I start mine in 4x6x2 inch containers. You can use recycled berry or mushroom plastic containers or any shallow container. It is easier to control the moisture level in shallower pots. When they have their second set of true leaves I transplant into 16 oz cups. If you want to go individually use dixie cups with a couple of seeds and then transplant into the 16 oz. cups. All containers must have holes for drainage. I have a crafters soldering iron that I use punch holes in mine.

Planting out---Find out the last frost date for your area and add a week or so. Your seed package will tell you how much earlier you have to plant before your last frost date--EG. if your last frost date is May 1st and a package says 6-8 weeks, you would start them 8 weeks before May the first.
You have to harden off plants grown indoors so they get used to the UV rays. For at least a week before you set your plants outside. You start by putting them out in a shady place for a few hours and then taking them back inside. Gradually you give them more time outside and more sunshine until they spend all day outside in the sun. Them they are ready to plant in your garden.

A hint--when you transplant your tomatoes sink the stem into the soil up to the leaves. The plant will grow roots all along the stem. Just be sure when you plant them outside they do not go too low in the soil that the roots will be in cool soil

I hope this helps and you grow beautiful plants next year

Your plants need 15 hours of light a day--a timer is a good way to go.