What's the best way to transplant my young plants?
First, make sure your plants are hardened off (see the Vegetable FAQ on Hardening Off Transplants). Also make sure they are well watered before you transplant so they're completely hydrated before the move. This will reduce shock.
Literally submerge the pots in a bucket of water, wait until there are no longer air bubbles coming up, and then sit them in another bowl for at least 20 minutes. Your plants will thank you.
Transplant in the evening, on a cloudy day, in the shade, or before a rain. Avoid transplanting in full sun or windy days which creates optimum stress on plants.
Know your soil. If they're going into the ground, get your soil tested (info from your County Extension Service) so you know exactly what your soil needs. It can be free or is usually in the $6 - $12 range. That's cheap! You won't be guessing at what to add to your soil and buying things you don't need. You can save money by doing this and you won't have to do it again for 3-5 years. Visit Soil, Compost, Mulch Forum so you know to till and amend your soil in the fall and not the day before you're putting your plants in when spring arrives.
If you're buying soil, go ahead and spend 7-10 dollars on a big, huge fluffy bag (around 15 pounds) of nutrient rich soil that doesn't feel chunky and skip the fertilizers. Look around at all the vegetation on the planet which grow just fine without humans adding fertilizer. Stick to homemade compost and you'll be doing yourself a favor in the long run. Fertilizers are a quick fix, but not good for your soil (or your pocket book) over time.
If you feel you must use fertilizer, go easy! Use half the recommended rate. You can always add more later. Consider that humans can take one tiny pill from a doctor which can make us feel better... a little goes a long way. Children can get sick eating too many vitamins... more is not better. If you have healthy soil, you don't need fertilizer. But if you still feel the need, then consider a treat of compost tea, fish emulsion, seaweed emulsion, etc a few days after they've adjusted to their new home. And again, not too much!
When transplanting, be gentle with the root system. Try to maintain the integrity of the soil structure to prevent shock, better safe than sorry (you experienced gardeners know the tolerance - this is for newbies). If you've got a four pack, don't take them all out and leave the roots sitting out in the sun or wind. Transplant one at a time. If you're planting into pots, keep the pots in the shade for a couple of days so your plants can adjust to their new environment.
Water immediately after transplanting. This will help the roots settle into their new home.
Have fun! Sometimes more problems come from humans trying too hard then letting the plant do what nature intended