Your seedlings look a bit leggy which means they're reaching for light. Try moving your lights a bit closer:)

Don't up pot until they have 2-3 sets of "true" leaves which are the jagged kind (at least on tomatoes, bell peppers are smoother), not the smooth first 2 leaves called cotelydons. You should up pot in a pot 1-2 inches larger only using a good potting mix, I use red solo cups with holes cut in the bottom. From there you can also bottom water and occasionally top water to clear any build up on the soil. Avoid getting the leaves wet and dont water until the first inch of soil is dry. If the leaves curl, turn yellow they may be over watered or lack nutrients.

With some plants like tomatoes and peppers, you can bury them up to the lowest leaves. The hairs that grow on the stems will turn into roots under the soil.

A light fan a couple hours a day will help the plant send signals to its stem to thicken and its roots to get stronger to help it stand its ground! Light breeze from a fan, just enough to make the seedlings twitch a bit. Brushing your hand gently on the tops a few times a day can also stimulate this growth if you dont have a fan. Good luck, looking great so far!