Being as our growing season is short, I start my annuals, perennials, and some vegetables from seed under grow-lights in the basement. This is my setup:
Heavy-duty metal shelving unit from Costco. The heights of the shelves are determined when you put it together. It is good to have some shorter and taller spaces for plants.
Black rubber floor mats to line each shelf, so that water and soil don't fall into the lights below. We bought the kind that comes from a large bulk roll and we cut it to fit.
Heating mat to provide bottom heat for germination of some seeds and growth of seedlings. These can be purchased in garden centers and online.
Fluorescent "shop light" fixtures. Each can hold two 48-inch fluorescent lights. Can be purchased from many department stores. I put two fixtures on each shelf (4 lights on each shelf). They come with little chains to suspend them and I adjust the chains depending on how close I want the light to be to the plants.
Fluorescent lights. In each fixture, I put one cool white (the regular type fluorescent that you would use in a house) and one grow light (is a little more expensive). The combination of the two provides a spectrum of light that promotes both growth and flowering.
Oscillating fan. I use an ordinary household fan to blow gently on the young plants. This really cuts down on mildew and fungus problems and encourages hardy growth.
Seed-starting mix. I buy bags of sterilized soil-less growing medium at a local store. It is also good to have some vermiculite or perlite. This is sprinkled ontop the seed.
Timer. This can be programmed to turn the lights and heating mat on and off.
Mini-greenhouse propagation kits. I have a wide variety of seed-starting supplies. I like the large trays with fitted clear plastic domes to maintain moisture for seedlings. I start some plants in plastic pots and others in peat pots.
Perennials are generally germinated en masse in a single pot in a sealed clear bag, and pricked out into individual 3 1/2" plastic pots. This ensures that you don't have half of your pots without plants, should germination be difficult. Seeds often need high humidity to germinate.
"No damp" chemical mixed up in a spray bottle to prevent "damping off" (death by fungus) of seedlings.
Yellow sticky traps for fungus gnats and other pesky flies.
Plastic label stakes so you can remember what you planted.
UV resistant permanent ink pen to write on the plastic stakes.