You dont have a lot on your list that cant grow well together. The hot and sweet peppers could cross-pollinate and surprise you, but Ive never had any trouble with that. Your peppers, tomatoes, carrots, cabbage, Brussels sprouts, and broccoli are long-season crops that will take a long time to mature probably all summer. The snow peas are a very early crop and should be planted out early in the season. Exactly when to plant will depend on the date of the last frost in your area. Other gardeners or your county extension service can help with that if you dont already know.
As far as direct seeding vs. transplants, snow peas, carrots, leaf lettuce, dill, coriander, marigolds, zinnias, basil, and the beans will do just fine being direct seeded in the garden. The rest should be set out as transplants. You can start them inside yourself or buy plants at your local garden center or big box stores.
Weve heard that veggie gardening is going to be really, really popular this year, so if youre looking for transplants, shop early.
For information on growing each of the plants youve mentioned, go over to www.bhg.com/gardening/plant-dictionary or click on the Plant Encyclopedia button on any page in the new garden channel and seach for each one.
Thanks for writing.