Saving seeds for many vegetables is not hard. You need to start with open pollinated varieties and not hybrid varieties. "Heirloom" varieties are a good way to start. I've done ok with beans, peas, and oats as well as different kinds of flowers.
Doing it well so that you can have a never-ending supply of your own seeds is harder. The main issue is purity -- keeping your desired varieties from crossing with other varieties (except where you want to breed your own, but that's a whole separate discussion).
Different plants are pollinated in different ways (wind, insects, etc) and have different tendencies to cross-pollinate with other plants. Some vegetables need hardly any isolation -- as @winwaed says in his answer, beans can often be kept pure without making any attempt at isolating them from another variety (as long as you don't grow two varieties side by side).
On the other hand, corn and tomatoes (for example) are very likely to cross. If you wanted to save seed from these, and wanted to maintain a pure variety (without random crossing), you'd need to isolate the plants (or at least the blossoms) so they aren't crossed by an insect. Some people will put bags on corn, for example, and manually pollinate the corn then keep the bag on to keep it from getting wind pollinated by another plant. I've intended to use the bagging technique in the past for both tomatoes and squash but end up getting caught up in the madness of harvesting and weeding and forgetting to do it.
Some vegetables are difficult to save seed from, especially if you live in a cold climate. Biennials don't set seed until their second year of growth, and most vegetables will winter-kill if left in the garden over a cold winter. Carrots, for example, are a biennial, and will usually rot (at least here in Zone 5b) before they can regrow and send up a seed stalk.
For some vegetables, there are special techniques to harvesting seed. With tomatoes, for example, you have to put them in a cup and let them ferment for a bit.
The go-to guide for collecting vegetable seed is "Seed to Seed" by Suzanne Ashworth. It has some general / introductory seed saving info in the beginning, and then a species-by-species reference that makes up the bulk of the book. It has considerations for the minimum population size needed to ensure genetic diversity (ie avoiding too much inbreeding), isolation distances, growing, and cleaning techniques for almost any vegetable you might find in a U.S. garden.
An additional book of interest is "Gardening When It Counts" by Steve Solomon. It is a general gardening book, but includes a section near the back of the book with information on a limited number of common garden vegetables and some practical advice on saving seed from each of them. He also debunks the pair of ideas that you can't start seed-saving with hybrid varieties and that "heirloom" varieties are a good choice for the novice seed-saver. (He's the original founder of Territorial Seed Company, so I'm willing to listen to his advice on this topic.)