They're much easier to handle after they've developed true leaves and have put on a bit of size. Then, just water the container well, and remove the soil block from the pot. Then gently push apart the soil block to separate the seedlings without breaking the roots. Don't worry too much about damaging the seedlings - they're actually very tough.
When separating the seedlings, hold the little seedlings by the leaf (NEVER THE STEMS) then make a hole in the soil in the new pot and gently guide using a pencil or pen to help drop it into the hole you made, gently knock / tap the soil around the seedling, place on a tray filled with water to allow the soil / seed compost to soak up enough water to allow the soil turn darker brown, then drain the excess water and place on another tray in a light sheltered place NOT direct sunlight, it's too strong from little seedlings, IF placing on a windowsill, I use an old newspaper to shelter the trays from the direct sun, make sure you turn the seedlings every couple of days or they will list to the one side to reach the light.
Have fun, don't kill the little plants with kindness but just good care. enjoy !!!!.
There's rarely a single way that simple things like this must be done... It works also to moisten the soil as you put it in the pots by direct watering and press it down lightly with your hand (rather than bottom watering). I also poke a hole for the seedling roots to go in with my finger, but I just fill in with moist soil with my hands - that way you have moist soil to an even level in the pot and firmed-up soil for the roots to grow into. Plant the seedlings to around about the same level they were at in the soil before - don't sweat this overly much though. You should be able to replace the potted-up seedlings into whatever light conditions they were growing initially (e.g. under lights, windowsill) - but if they were outside, I agree with setting them in the shade for a while until they strengthen their root systems.
Whenever you move plants from inside to outside, you need to first harden them off (setting them in the shade outdoors for 5 days or so works fine here).