If you're curious about the difference between heirloom seeds and the hybrids you get in big garden centers, I wrote a little about it here:

How to Interplant to Get the Most Out of a Small Vegetable Garden

How to Save and Store Heirloom Seeds

Gather Seeds From Each Plant

Some plants are easier to harvest seeds from than others. For example, bell peppers seeds are extremely easy to gather, while lettuce seeds can be more difficult. The seeds should come from your strongest plant, and from fruits or veggies that have been allowed to ripen fully on the plant.

Here's a quick list of the seeds you can gather and how to do so:

Wet seeds (cucumber, melons, squash): Allow the fruit or veggie to ripen fully on the plant, then pick, cut, and scoop the seed into a large bowl. Flood the bowl with water and allow the heavy, viable seed to fall and the dead seed and pulp to float. Pour off the top and repeat 2-3 more times. You can use your fingers to gently separate the seeds from the pulp if needed.

Tomato Seeds: Tomato seeds in particular need to be dealt with differently. You will need to scoop the seed and pulp into a small cup or jar and add some water. Let this sit and ferment for a few days to a week (preferably outside because of the smell) until a white mold has formed on top for about a day. Once that has happened, clean them just as you would wet seed.

Dry seeds (corn, beans, peas, peppers): You will generally let these dry on the vine or stalk, pick them, and then crumble any pods and separate the seeds.

Bulbs (potatoes, garlic): When harvesting potatoes, choose the smaller specimens with at least 2-3 eyes to save for seed. Do not wash them! Just store in a cool, dry place (around 50° F) until 4 weeks before planting, and then sprout as usual. For garlic, save healthy bulbs and replant individual cloves separately.

Seeds from plants without obvious seeds (leafy greens, beets, carrots, celery, broccoli, onion, herbs): Allow these plants to overwinter and sprout their seeds. Let them dry on the plant and then harvest.

Dry Your seeds

You want to get your wet seeds as dry as possible. For this, you can use cheapie paper plates. They absorb excess water and won't stick to your seeds as badly as paper towels once they're dry. If you have a ton of ceramic plates and saucers, you can dry them on these and they won't stick at all, but most people don't have that much dishware to devote to drying seed.

Place each variety of seed on a different paper plate and label with a marker. You want to leave your seeds for a few weeks to make sure they dry thoroughly. You don't want moldy seed! 

To help keep them from sticking to the paper plate as they dry, you can stir them daily with your finger until they're dry enough not to stick.

Put Dried Seeds In Labeled Envelops and Store in a Cool, Dry Place

After your seeds have dried thoroughly, you can place them in envelopes labeled with the seed variety and store in a dark, cool, dry place. You could store them in a shoebox in the back of a cool closet and they will keep until next year, or in your freezer for several years. 

You want to put the date that the seeds were harvested, any planting directions, and any other information you need to remember. I came up with a printable seed packet and this is what I use to store my seed in. You can get that here:

Heirloom varieties are some of the tastiest and easiest to grow and save. If you already have heirloom varieties in your garden, you're good to go. If not, purchase heirloom varieties from your local farmer's market and harvest the seeds so you can have the best plants in your garden next year. You can also find heirloom seeds online by searching "Buy Heirloom Seeds." There are many companies that specialize in them, and they will pay for themselves pretty quickly!