Mind you - I'm NOT an expert!!! But, here's what I do...
I start my seeds in rows in flats. The lights go about 2 - 3 inches above the soil. Mine stay on 24/7 until each plant is happily in its own container, then its off and the plants are in a sunny window (which requires turning them daily, usually - so that they don't grow all to one side - they will 'follow' the sun.)
Keep all seedlings moist, but not wet. They shouldn't damp off that way - at least if you use soil-less potting mix or sterilized soil. Keep the lights as low as you can without touching any plants... The higher the lights, the more spindly the plants. Plants grow to light - any light... Keep it low, and they are fuller, greener and healthier.
The seedlings are transplanted as soon as they get their first set of true leaves... This means that you have a pair (usually, a pair, sometimes only one leaf, though) that resembles what the parent plant will look like. True leaves are the first 'adult' leaves, and the second actual set that forms. The first leaves you will see before the 'true' ones usually are rather round and non-descript. This is also when they get their first dose if you fertilize... A weak manure tea would be good... but I cheat and use Miracle-gro - either the soil (so no extra fertilizers!!) or the mix - weak.
If you use the peat pots, you only plant one seed in each, so its really easy to skip the waiting - you just plant the peat pot as per the instructions in the other post.
If you use a flat, use a table fork to pick out your seedlings, or, as I do, pinch them up with a little of the soil around the roots.
Seedlings are pretty hardy for their size, but do be careful to keep the roots intact as possible (nearly impossible not to lose a few) and not damage the stem. For tomatoes, replant into pots deeper than they were... Their stems will root in moist soil, giving you a better and heartier plant. I usually do it with everything, but there are better experts on particular plants than I am, so....
If you leave them in the flats too long, you will see roots coming up through the top of the soil, and they will be hard to seperate, but don't despair. Under normal conditions, you will never have to add more soil to the flat - just transplant into their first new containers. You cannot avoid losing some roots when this happens, but the plants will be ok. You might have to break some of the roots at this stage, but keep as many as you can attached to the plant. Some things, like tomatoes (again!) may be transplanted into larger containers 2 - 3 - even 4 times before they get hardened off and put in the garden. Gram's used to do that - I just start mine a little later, lol.
Hardening off - I count my first step when I put them into individual containers and set before the window. After a couple days there, I open the windows for a couple hours a day... Then, on really nice days, I take them outside for a couple hours, leaving them a little longer each day until they can stay out overnight. Once they've been staying out overnight for a few days, then they go into their spot in the garden. After being pampered so thoroughly under the lights, they need the gradual hardening off process to become accustomed to things like sun, wind and sprinkles.
Good luck!!!
Sue