from seeds. Because I never have, until this year. I saved some seeds from some Green Zebra tomatoes I had last year, put them in egg cartons with potting soil, and now they have sprouted.
Help!
Mine have always done terribly, so I’ll just offer moral support and wait for someone who knows what they are doing to come by. 
Well depends on your location. In general, 'maters like full sun, warm temperatures, and lots of water. Plant your seedlings in a rich rich mix of compost, manure, and earth; water well, and trellis them up as soon as you can. Always encourage upward growth and give them lots to support them.
Where are you?
Don’t do anything till they’re a pretty good size, like the started plants in the nursery at the store.
If you used a paper egg carton, you can just plant the whole thing. But paper is rare, so you need to gently coax them out and put them in the ground to just below the first leaves. How close you put them is up to you, but you need to keep in mind how large thet’ll grow. This varies. Also, if you have plenty of seedlings, you’ll want to put two or three in one spot so you can thin them when they get established and sacrifice the weaker ones. These culls are not good to eat. They’re members of the nightshade family.
There’ll be a hundred or so more opinions here on how to do this soon, so sift through and pick the methods that appeal to you. We’re all experts, you see. 
I saw your other thread about lights, OO, and posted there, too. Upshot, keep the lights about an inch above the seedlings.
Assuming your seedlings are in styrofoam egg cartons, make sure you poke some holes in the bottoms, so there is drainage.
Tomato seedlings are prone to damping off, so when you water, try as much as is possible not to get water on the seedlings, just on the soil. If you’ve got several seedlings in one “cup,” (10 seedlings, 1 cup?!??), thin them so you have at most 2, and those should be as far apart as they can be.
When your seedlings have a couple of sets of true leaves, you will want to transplant them into something larger. If you’re into reusing things, a yogurt container with holes poked in the bottom would probably be a good size. You don’t want it hugely larger, because that makes it easy to overwater.
I used to make pots out of newspaper. Then, when the time came, I planted the maters pot and all.
You probably can tear the egg carton apart to free the seedlings. The goal is to disturb the roots as little as possible. If you have to separate seedlings, use a fork or something to very gently tease them apart. Put seed starting mix, not potting soil, in the containers. Potting soil is another common cause of overwatering.
As your plants continue to grow, you may want to transfer them again, to a 3" or 4" pot.
You’ll need to harden your plants before you put them in the garden. A week or so before you intend to plant, set them outside in a shaded location. If the weather is really mild and you know the plants will be in shade ALL the time, you can leave them there. Otherwise, you should bring them in when the sun starts to hit them or if the weather turns cold.
Tomatoes like the soil to be well warmed. What zone are you in? IMO, you will gain nothing by planting them early and trying to protect them with Wall-O-Waters or similar. Just wait till the soil is good and warm and the nights are in the 50s. I’m in Zone 5 and I never plant tomatoes before the end of May, beginning of June.
Good luck! Growing your own from seed is FUN!
Go to the homepage of the site you linked and sign up. They offer advice and to sell you more stuff. 
Really, it’s generally a good idea.
mangeorge
What freckafree said.
I love this site, Lehman’s.
Jiffy Peat. Water. Love.
If you want more specifics, I’m here. It’s what I do every year at this time. If what’s at the other site is OK, I’m cool with that too.
Don’t forget tapping the mature plants gently with a stick every day when they begin to flower.
Heh. I make my own newspaper seedling pots. Figures someone smarter than me would figure out a way to get $13 plus shipping out of people. BTW, newspaper doesn’t break down that fast, so you might consider just gently ripping the newspaper away from the rootball before transplanting.
Ok. I give. I’ve been raised on a farm all my life, and I’ve never hit a tomato plant yet. Why would you do that? 
I did rip this guy off the plant and jump and down on him. 
It’s a cool gadget. Cost is not a consideration. You will comply.
I’ve used newspaper, maybe two layers, and the roots poked right through. But your point is a good one. The paper comes off very easily.
Story is that it helps the flowers set, by stressing them slightly. I’m sure I’m not the only one who’s heard it. Don’t know if I believe it.
It probably has to do with the way tomatoes self-pollinate. I tried buzz pollinating last year and got pretty good results.
My daughters used to have contests to see who could hold on to one of those the longest. It does feel weird. :eek:
They’re very easy to control, but a little hard to see. Just a toss will do it. My neighbor’s tool shed had these “mystery spots” all over it. 
The chickens would always follow me around as I checked for those little buggers. They would grab one and run like heck, with everyone else trying to get it from them. Apparently they’re like potato chips…for chickens. 
To answer a couple of points: It looks like I am Zone 5a/5b. Planting is about 5 weeks away. I am using styrofoam cartons. I use the top to hold a little water, and poked holes in the bottom part for the water to wick up into the cups. I have about a dozen nice-looking seedlings, and about an equal number that are way too leggy. Will they stay small enough to keep them in the cartons for another 5 weeks? Or do I need to start building paper pots? (Great idea, by the way. I’ll see if I can putter around and come up with something.)
I meant to ask: how much light (hours per day)?
If you’re in Zone 5a/b, you’re WAY more than 5 weeks away from planting tomatoes. Yes, the frost-free date is supposed to be April 30, but trust me on this. If the soil isn’t warm, your tomatoes will sit in the ground and sulk. Last week of May, at the earliest.
I put my lights on a timer – 12 hours on, 12 off.
Here’s a pretty good video on making newspaper pots. I didn’t listen to it, so I don’t know how annoying she is in that respect.