First thing I do is soak the punnet, briefly in a bucket of very mild seaweed solution. (Maybe a cap to a 5 litre bucket). This makes them easy to take out. Make some holes for them and then squeeze the base of the punnet. NEVER grab a seedling by its stem, if you squeeze it won't be able to daw up water and nutrients and will die. Pop the seedling into the ground and pack in firmly around it. Then water with the seaweed solution, carefully and
re pack if the water has loosened things up. As you give them this watering you say the magic words: (No I'm not kidding, I do this every time and I feel very weird if I don't do it). Where was I? Yes, say the magic words: "Grow little plant. Grow, grow". So after a punnet or so this is how things look.
These are spinach seedlings I put in yesterday.
OK, you're not finished yet. The first 48 hours are critical and so you want to protect your seedlings from harsh sun, wind, rain etc. To do this you add covers. Now I find old large yoghurt containers work a treat. You pop them over the seedlings and leave them for two days. No, they won't die, they will thrive.