I sowed my tomato seeds a few weeks ago and they're strong enough to move (see my blog, "Growing Tomatoes From Seed"). The first set of leaves that appear on a new sprout are the cotyledon leaves. They begin the photosynthesis process and get the plant growing. Typically they're oval shaped and don't look like the leaves of the adult plant; that's why it helps to label your seeds so you know what's growing. Soon after, true leaves develop. At that point there is enough root growth to support the plant and avoid the stress of transplanting. Wait for one or two sets of true leaves before transplanting.
My tomato seeds were sowed in small peat pots. When done properly you can expect all or nearly all of your seedlings to survive the transplanting process. Just for extra insurance, I leave some of the seedlings undisturbed in the original pot while I move the overcrowded ones to their own new pots. After the danger of frost I'll transplant all of them into the garden.
Using a fork, a chopstick, a spoon, a popsicle stick, a knife, or any other small utensil, gently dig out out the seedling or group of seedlings that you are ready to move. Try to dig deep enough that you don't disturb too many of the young roots. Digging out the seedling with a clump of soil attached is fine.
That's all there is to it. Transplanting seedlings will double or triple the amount of space required to store them, maybe more. Anticipate where you'll put the new pots before you begin the process. If you don't have adequate light for all seedlings, it may be better to leave them in the original pot and just thin out the smaller seedlings. That's one reason I recommended starting seeds in pots and avoiding transplanting.
With proper water and light all of the seedlings will continue to grow and be strong enough for the garden soon. I'll cover that step in a few weeks.