Welcome to ATP, Linda!
My biggest problem when I first started out transplanting seedlings was compaction. Three-fold in my case; pressing the mix against the root mass, shaking too much to settle the mix, and over-watering from the top, especially right after I set the seedling in the new container. They'll take off much better and faster if you fill your new container with lightly pre-moistened mix, toss the seedling in, and then just drip a few drops of water between the old transplant clump of mix and the new. Then set the container(s) in a tub or dishpan with warm-ish water and let it soak for 5 or 10 minutes. Remove, allow to drain and you're done.
Let your babies' mix dry a bit between waterings. A slightly droopy transplant is okay for a day or three. (I used to panic at this point and proceeded to half-drown the poor things.) Drier mix yields bigger and better plants in the long run, while soggy, always-wet soil will produce weak ones.
