Seed Starting

I start many vegetable and flower plants from seed indoors. Indoor seed starting is easy to do, saves money over buying transplants, and it also allows me to start "gardening" in late winter, when I can't wait to start playing with dirt again! The picture below shows one of my flats with coleus, amaranth, cutting celery, convulvus tricolor, Thai and Genovese basil, tomatoes, and parsley from this spring.

Seedlings growing in Jiffy-7's and fiber pots

Containers

There are many different types of containers and growing media that can be used for seed starting. Debates over which method is best can get heated, but in my experience as long as you have enough light, water, and the correct temperature, plants will grow quite happily in pretty much any container and growing medium. I personally like to use Jiffy-7's to start my seeds, because I find that they save me time. Jiffy-7's are small discs of compressed peat, encased in a small mesh bag. You soak the peat discs in warm water for about 30 minutes, and they expand into small cylinders. I like these little guys because:

1. It's difficult to mess up the watering with peat. Peat is a very absorbent material that will hold a lot of water, while also allowing plenty of oxygen between the fibers. Despite the fact that the peat cylinders are tiny, you can water only once a week or so. The containers will not dry up suddenly without warning, and there is also little chance of the roots drowning due to overwatering.
2. The plants' roots grow right through the mesh bags. If you use a container with hard, impermeable sides (like plastic), when the roots reach the edge of the container they will begin to circle, which is not good (they will continue growing along this circular trajectory even when freed from their container). You can prevent this by being super-vigilant and transplanting to a larger container before it happens (totally unrealistic for me!), or by scoring the roots after you take them out (can get to be a lot of work). With a root-permeable container this is not an issue. The roots of the plants all grow into each other, and some roots will inevitably be torn when you remove one from the flat. This is perfectly fine - a bit of root pruning encourages new growth.
3. No need to fill with potting soil. Just plant right into the cylinders. Eliminating the soil filling step saves an amazing amount of time when you're planting a lot of containers.
4. Transplanting is quick and easy. Just pop the entire container right into the garden soil. Planting out a large flat goes very quickly when you don't have to remove each plant from its individual container.

Marigold seedling with roots growing through container

You can also achieve great results by planting in large open flats with no divisions, or using homemade newspaper pots or individual peat pots filled with potting soil. But these options will mean a bit more work.
 
Lighting

Vegetable and most annual flower seedlings need at least 11 hours of bright, direct sunlight a day. It is impossible to get the necessary amount of light with even a bright south-facing window. Without supplemental lighting your seedlings will be leggy and weak. I have a simple setup with LED grow lights and metal grid shelving. Fluorescent grow lights can also be used. Fluorescents are cheaper to buy initially, but use more energy and are thus more expensive to run than LEDs. (I'm not sure which one ends up being cheaper in the long run, but both work.) Fluorescents get very hot, so must be kept further from the plants than LEDs. The LEDs emit a weird-looking purple glow (my neighbors must think I'm doing some kind of mad scientist), but the plants seem to like them. You want to position your plants so they will receive as much natural light as possible, in addition to your artificial lighting. I have mine in my sunniest window, and also keep the lights on for 11 hours per day.

Seedling flats growing under LED lights

Transplanting

Once the roots start growing out through the sides of the containers, it's time to either transplant out into the garden or into a larger pot. I use fiber pots which are also root permeable for this (see first picture above right), filled with any kind of potting mix amended with organic fertilizer. Before your plants go into the garden, they need to be hardened off so they get accustomed to the bright light and buffeting winds of the outside world. I harden off opportunistically - as soon as the seeds have germinated, any sunny afternoon above 50 F they spend a few hours on my deck. I always bring them in at night. Because they've spent so much time outdoors from a young age, my seedlings don't really need a strict "hardening off" schedule before going out.

Seedlings transplanted into garden - early April

I transplant tomato and pepper seedlings out early using Kozy Coats (the red thingies), but broccoli, lettuce, snapdragons, and parsley can go out unprotected rather early. For basil, marigolds, zinnias, cucumber, and squash wait until the danger of frost has passed. Peas, spinach, radish, carrots, beets, and beans are better direct seeded.

Lettuce 'Deer Tongue'

Before you know it, you'll be harvesting yummy veggies and enjoying an early flower display!

Marigold 'Favorite Red' with Basil