Do you have different varieties of the same vegetables? Do you have hybrids?
Many different types of vegetables will cross-pollinate with other varities of the same vegetable. You could plant the seeds, but you never know what you'll come up with. For instance if you have a "Big Boy" Tomato plant and a "Celebrity" tomato plant, they will likely cross pollinate one another. The resulting seed would be neither a Celebrity or a Big Boy, it would be a mutt. Possibly a mutt with undesireable characteristics.
However, if you have two Celebrity plants, and your neighbors aren't growing tomatoes too close by, then saving them would result in Celebrity seed.
If you're growing only one variety of each plant, saving seeds is a cinch, and would work with all the veggies you mentioned.
HOWEVER, You can't save any kind of hybrid seeds that will be pure. That's one great reason to only plant open pollinated plants.
Your peas will not usually cross, so saving them should work fine, regardless of how many varieties you planted. Let the pods dry out completely on the plant and store them in a cool dry place. I keep all my seeds in a cabinet in my room. You can refridgerate seeds for longer life but it usually isn't necessary.
Corn will readily cross unless you only have one variety. Same with the peppers.
If you only have one open-pollinated variety of each, let the Corn dry completely on the plant (after the whole plant turns brown) and then pick off the kernels. This is your seed. For the peppers, let a few of them turn red on the plant and then dig the seeds out and let them dry on a paper towel, but not in direct sunlight.
Zucchini and Summer Squash will cross with other varieties AND each other. So unless you hand-pollinate you won't be able to use these seeds
If you are interested in saving seed, I highly recommend the book 'Seed to Seed' by Suzanne Ashworth. She gives details on saving seed of every vegetable plant imaginable, even if you do have multiple varieties.
Seed saving is a great way to save a money and keeps old varieties alive that may be in danger of extinction. If you decide to do it, then good for you! But it can be very confusing to start with. Hope this helps!