Should I just buy a 48" shop light? I have two 10 x 20 flats. I've never done seeds indoors. I have about 6 weeks until our last frost date, May 15 in Chicago. I already started the seeds on a heat mat with domes. I didn't realize they sprouted already. Now I am in a panic. The grow light I bought seems to be too small. 18W Grow Light. I'd like to buy something today at a local big store.
This site says you need 50 Watts per square foot of grow area. So, just calculate your required Wattage, and then purchase a grow bulb of the required size.
I grow my tomatoes under two 48" shop lights, with no special bulbs (they need the extra light the wider spread of two lights provide). I hang the lights from my basement ceiling with chains, which I raise as the tomatoes grow. As noted in my comments, I repot the seedlings into deeper pots as they grow - generally from normal seed-starting cells into 4" pots, then 1 quart (a deeper 4" pot) and maybe a gallon pot, each time, burying them deeper.
I don't usually start my seedlings until the second week or so of April, though (in Wisconsin) - you're a bit early :)
If the plants are etiolated when May 15th comes, just bury them deeper in your garden. Some experts say to not remove any leaves, but I've found that you may have to if the plants are too leggy.
I ended up getting four 48" double bulb shop lights. I started with 10 flats on heat mats. Transplanted into 3 or 4 inch pots, then 6 inch, and hardened them off outside about a week before I planted them out. It didn't seem to mater which bulbs I used. I tried different kinds the next summer. It Worked great.
Your biggest problem will be soil too warm after sprouting , this will make seedling too tall/leggy. Long ago I grew many plants from seed in Chicago area. Once sprouted , I moved the flats to a garage that was usually 50 to 60 F. I used 48" shop lights with a mix of grolux and white bulbs but I added 2 incandescent sockets to each hood with 60 watt bulbs ( there were no LED or fluorescent bulbs then ). I put the lights very close to the plants, occasionally leaves burned on the incandescents but that did not slow the growth. Today , I would use LED lights and would not need incandescents . Put the lights very close to the plants. When it was probably not going to freeze anymore, I just planted them outside : Except heat lovers like tomatoes, peppers and eggplants , I would keep then under lights until late May.