I planted a number of vegetable seeds (zucchini, cucumber, peppers, kale, etc.) indoors in 1"x1" pods. They germinated quickly and well. Once the true leaves appeared, I moved them over to a 4"x4" container with potting mix. The plants look healthy but they are not growing as fast as I would have thought. I now leave them outdoors (in the shade) and have been out there for about 4 weeks. What else should I be doing so that they can grow and I can move them to my vegetable raised bed?
I'm by no means an expert, however I am pretty sure that, at least for Zucchini, Cucumber and peppers 1% more light = 1% more growth. Thus in order to increase growth, maybe you should start moving these into full sun.
You need to harden your seedlings now so that they are ready to be transplanted into your garden bed. Since your pots are on the small side, there's a risk for some plants that they'll get the signal to start bolting.
This video from the biointensive grow method explains the transplanting process. They advise using 6 inch deep flats, wait until the plant is 6 inches high, and transplanting in a hexagonal pattern to maximize soil use.
Seedlings (from my observations in my area) don't tend to grow very fast outside in containers in the spring, when exposed to the raw outdoor air and light. They seem to require more warmth. I don't know how warm your area is, though.
However, if you can put a humidity dome over them outdoors, that should help them to grow faster (provided there's nothing wrong with the plants or the soil). You can make a humidity dome out of an empty milk jug. Throw the cap away (air and circulation is important), cut the bottom off, and put it over your plants (make sure wind won't blow it away). Plants grow faster under them until it gets too hot to use them without killing them. When it's hot enough to germinate cucurbits via direct-seeding, you should probably think about removing the humidity dome soon, if not right away (or at least widening the hole to let more air in and prevent the sun from cooking the plant).
I suppose it's possible that the plastic of the milk jugs filtering out UV rays would help the plants to grow faster. The increased warmth would certainly help (the jugs will even protect from frost). The protection from wind and pests would also likely help.
Kale should be able to handle cool temperatures better than the other kinds of plants you mentioned and so should be in less need of a humidity dome.