Saving seeds is a great, economical way to produce the same veggies you love without having to purchase seeds. The veggies must be open pollinated; hybrid varieties will not work. In most cases, it is very easy to save seed, although sometimes it can get a bit complicated. Some easy veggies to save seed from are peas, beans, peppers, and tomatoes. Peas and beans are actually the seeds themselves. In these cases, you just let several pods stay on the plant and dry until the seeds inside are hard. Then remove the dried seeds from the pod and store. Pea and bean seeds can last as long as 8 years. Seeds saved from peppers and tomatoes should be saved from the best fruit. Pepper seeds just need to be stripped from the inside of the pepper and dried. Tomato seeds must be fermented to break down the gel that coats them, good step by step directions can be found here. After they have been fermented, they are simply dried and stored.
In order to save seeds from most other plants, you have to let the plant go to seed. This is when the plant starts to produce flowers, which then become pollinated, and then form the seeds. We have done this with herbs and are currently letting lettuce and broccoli go to seed. You should only let one variety of each vegetable go to seed at a time, since they are setting flowers they can be easily cross-pollinated. We let the formed seed pods dry on the plant, then harvest the seeds and store. The International Seed Saving Institute is a great resource for information on saving seed.