I'm in my second year of trying a vegetable garden, and last year I started planting way too late. This year, I want to start some plants indoors so that I can transplant them outdoors earlier. I've saved some styrofoam egg cartons and read a bunch on how to start the seeds in them. But now I'm reading that a lot of people only start them in egg cartons until they sprout leaves, and then move them to bigger indoor containers before moving them outdoors. We're getting close to outdoor planting season here. Do I really need to move them multiple times, or can I just start them in the egg cartons and them move them out into the garden?
No, you need to move them into pots before putting outside, and they should be moved once they develop 2-4 sets of true leaves (not counting the first set, the cotyledons) at the latest. The plug of soil in your egg carton won't be big enough for proper root formation, and any plant is only as good as its roots. Move into small pots, grow them on a bit, then harden them off before planting outdoors in your prepared area. It's one of the tedious bits about growing from seed indoors, the need to pot on I'm afraid, and it's not a step you can omit. Starting with larger containers doesn't help for individual seeds, they're better started off in something shallow (your egg boxes or seed trays, peat starters, whatever) then transplanted as seedlings into something larger.
Depending where you are and what vegetables you've chosen, some seeds can be planted straight into the ground a bit later on, into a prepared area with friable soil.
I like what @Bamboo has said. Good information.
I would add that the goal here is to create a healthy plant with a well established root structure and leaves. I exclusively use soil blocks for seed starting now. I've done the plastic tray inserts, the yogurt cups, egg cartons, newspaper pots, etc. Those all work well enough. The plants - and you can see this directly with soil blocks because there's no container to block your view - push out roots quite readily once they start growing and the roots seek out moisture and nutrients. If they don't get those things, the seedlings fail.
I start all my seeds (that are not direct-seeded into the garden later) in either 3/4" or 2" blocks. The 3/4" blocks are moved into 2" blocks and the 2" blocks are moved into 3" pots - I just bought an entire sleeve of these, but I've also used those red plastic "Solo" cups too quite effectively - and the 3" pot is my "transition" pot for most everything I grow with the exception of tomatoes and peppers.
That time in the 3" pot gives the plants considerable space for further root growth before being put into the garden. It also gives me a bit of time to get the garden prepped.
Two things about transitioning plants outdoors - the "hardening off" period is really helpful in getting them accustomed to the temperature and the wind. I was skeptical early on that this was really necessary but I've seen much better results when easing them out into the outside world. Another thing is being mindful of the "last frost date" - better established plants can survive a light frost that sneaks up late in the season. Smaller, less established ones can be killed. It will depend upon the plant type, of course.
You could try your hand at a cold frame with some percentage of those seedlings and see how that goes. As long as you actively manage the heat and the wind (and keep them watered, of course) you might have some luck. I like to experiment a bit so that sort of things is something I would personally try just to see what happens.
In addition to the great advice above, if you have the opportunity to use cellulose based egg containers, then that would be better because you can plant those directly just like peat pots. The roots can grow right through the softened cellulose.
I like to use peat pots myself, they are a minor expense, and you can make your own from compost, or newspaper, etc. if you buy yourself a press.