One of the most rewarding things I do is to grow plants from seeds. After years of gardening, I'm still not callused to the miracle of such enormous potential (with vegetables, such enormous production) emanating from such a tiny casing.
Though I usually start with seeds I've purchased to make sure they're healthy and will yield what I'm looking for, in the last couple of years, I've saved a few seeds from wonderful crops just because I couldn't help it. If the crop was so good, I've reasoned, it's the seeds from those very plants that will ensure great success next year. In most cases I was right, though horticultural success isn't all I gained. A more intimate understanding of what it means to be instrumental in the succession of fruit and vegetable generations has brought me to deeply revere the fundamental sacredness of food in my life.
All that said, I suggest you start, as I did, with a couple of very easy vegetables to save seeds from: tomatoes and beans. Both are self-pollinating (which means you won't be saving seeds from a yet-undiscovered cross between your prized vegetable and an unknown neighbor), but you do have to be sure to save seeds from non-hybridized varieties (the seeds from hybrid varieties always revert to traits of one of the parents).
Though I've sliced a tomato, let the slices dry in my cold garage, then broken out the seeds to plant in the spring, that's not the most secure way to save tomato seeds. It's best to scoop out the gelatinous material that contains the seeds (do this by cutting open your most prized heirloom tomato when it's ripe-cut through it horizontally to expose the cavities that contain the seeds-and gently squeeze the half to press out the gelatinous stuff. What's left you can make into salsa.
Mix the gelatinous, seed-containing material with an equal portion of water, put a lid on it and let it stand for 1 week at room temperature. It'll look pretty disgusting (covered with mold and so forth), but that's right. Scrape off the mold, pour the contents into a sieve and rinse. Collect the seeds and dry them. Store in a cool, dry place.
Saving bean seeds is even easier. Simply let the pods dry on the plant, then harvest and shell them. Store the bean seeds (yes, that's the part we normally eat) as you would the tomato seeds-cool, dry place. RB