It is probably a little bit early given that the winter has been so long this year, but I am starting the tomatoes and some of the other more delicate plants now. Peas, beans, cucumber, melons, and squash are all a lot more robust and grow very quickly. These varieties will all be started from seed directly in the ground after the frost has passed sometime between March and April.
I'm using some left over containers from plants bought at the nursery last year. They are a little bit bigger than what you normally get at Home Depot, etc ... for starting plants, but there will be less to worry about in terms of the plants getting root bound or needing to move them to a bigger container later.
I'll plant 3 - 4 seeds per container and thin them to 1 plant per container once they sprout. There are 72 containers between the two trays.
The dirt is regular dirt from the garden which has been sifted and cleaned.
I keep them in the living room next to the big picture window. There is also a heat vent right there, so they will be warm at night in addition to getting the sunlight. The living room has the basement under it, so I do not need to worry about the floor getting too cold. Some people will put a heat pad under the seedings, but with this set up I do not think that is necessary. In previous years, I used to put the seedlings next to the patio door where they would get light, but the heat vent was not there and the floor was much colder. This is also a less trafficked area and once they seeds are planted, they will not require too much care other than keeping the watered.
Note also the cardboard under the planters to help keep dirt and water off the carpet.