I did a small quarantine experiment with some overripe grape tomatoes that I need some tips on. I managed to harvest the seeds and plant them and voila! Over 50 tomato sprouts. Now the question is...what is the next step with these spouts? Texas summer is coming and I hope they can survive. I am not sure if I should be repotting them so they aren't clustered together or start planting them directly into the soil. Tips appreciated!
You will need to prick them out into individual small pots and grow them on into small plants before planting outdoors in the ground. Select the strongest looking seedlings to transplant if you have too many, and some potting soil to grow them on, preferably a seed and cutting potting soil initially. As they grow, move into larger pots with ordinary potting soil, then plant outside when they are a good size,around 8 inches tall with sideshoots.
If you're growing them on indoors, you will need to harden the plants off before planting outside so they have a chance to acclimatize to the new, outdoor conditions.
There are a lot of ways to do this. There's no single best practice, in my opinion. Bamboo's answer is pretty good, except you should know that splitting them up at the roots is going to set them back, slow growth, and probably impact long-term production--this may not be as big of a deal for tiny tomatoes, or if you fertilize well, however. I've experimented with splitting them up versus just thinning to one plant, and there's a noticeable difference, IME. Undisturbed roots have better results. I still split some up sometimes, but I try to avoid it, and I try to be careful.
What I like to do is overseed (in containers in full sun outdoors, which I bring in at night when it's supposed to get below 42° F.), let a bunch of tomatoes sprout, grow them until they're 4+" tall, transplant them like that (after the primary risk of frost is over), and thin each to one plant (with scissors) maybe 8 to 10 days after the transplant.
Most people don't do this process, but there are a lot of advantages to holding off on the thinning until then (it's insurance against cutworms, pill bugs, the occasional dying plant, etc., and they don't fall over or get drowned as easily). Once thinned, they increase in size rapidly.
Because they stay outside most days (I keep them in on Sundays), they don't need to be hardened off.
If your area doesn't get much rain (rain leads to increased nitrates), you might try giving them foliar sprays of calcium nitrate weekly until they're strong (two or three weeks might be enough), starting three days after the transplant or so. Nitrates seem to help young plants mature better (including tomatoes), even when it's hot and dry. You might water them once every day or two, too, if they dry out—until they're stronger.
I think calcium nitrate can also help plants overcome being split up, too; it seemed to worked well with my peppers, this year (I split up a very few overseeded containers into about 42 plants at transplant time). Peppers are related to tomatoes, but they grow more conservatively, and have more sensitive roots.
FYI, don't give foliar sprays when the sun is shining much.