When the seedlings are large enough to handle, they can be transplanted or pricked out into bigger pots by lifting the seedlings individually, holding each by one of its true leaves.
Keep as much compost around the roots as possible during transplanting and use a dibber (or a plant label or stick) to make a hole for the plant in the new pot of compost.
Once the weather is warm and the plants have grown sufficiently that they are strong enough to be moved outside, hardening off will need to take place - the gradual acclimatising of the plant to the outside by moving it into increasingly cooler situations (for example from windowsill to greenhouse to sheltered part of the garden) or by simply putting the seedlings outside in semi-shade on nice warm days.
Commercially, seedlings are moved from a warm, humid propagation tunnel into a cooler polytunnel or glasshouse to harden off before pricking out and potting on.