Also important is to keep them in a bright spot so they don't go tall and leggy. If you don't have good light for your seedlings you can use grow lights. These have the right spectrum of light for plants which is different to ordinary light bulbs.
Seedlings that are kept indoors until the weather is warm enough for them like tomatoes might outgrow their small pots, plugs or seed trays after a few weeks. Once you see roots coming out at the bottom of the pot and the seedlings have set the second set of leaves it is time to transplant them into bigger pots.
We always add some organic fertilizer and seaweed powder for extra minerals to the potting soil. This will keep the seedlings growing healthily and vigorously and we don't have to worry about adding a liquid feed to the water for 5-6 weeks.
If you don't do this you will have to use a liquid organic fertilizer once a week. I would also recommend using liquid seaweed because it really helps the seedlings to grow a good root system and to be disease resistant.
More info on using seaweed in the garden here!
I have to admit that we often skip this step because we have just too many plants and seedlings. Instead, we use garden fleece to protect the young plants from too much sunshine and cool temperatures. I explain the benefits of using fleece here.
You can also use a cold frame or hoop tunnels for hardening off the seedlings.