I don't save vegetable seeds, but I do save some flower seeds that I use in large quantities. I just lay out the seeds and store in a paper bag or envelope in the garage.

Procedures for drying & storing seed will vary, depending upon
the type of seed, the storage environment, and how long you intend to
keep the seeds. Most seeds will keep for several years in paper
envelopes; but to get the longest storage possible, the seeds should be
stored either refrigerated or frozen, in a sealed moisture-proof
container. This is especially recommended for short-lived seeds, such as
onions.

Seeds that you save from your own garden should be
fully mature, and allowed to dry to a low moisture level before storage.
The amount of drying depends upon the type of seed, and how much
humidity is present in the drying environment. In dry areas, or periods
of low indoor humidity (such as dry Winter air), a week or two is
probably sufficient. If humidity is high, it may be necessary to use
some desiccant (in a sealed container) to dry the seeds.

Purchased seed should already be dried to the proper moisture level for storage,
unless they were exposed to moisture or a period of high humidity.

If you choose to use plastic bags for seed storage (I do), then I recommend using freezer storage bags. They are made of thicker, less permeable plastic, and have tighter seals.

When preparing to use seeds that were refrigerated or frozen, be sure to allow the container (and the seeds within) to reach room temperature before opening. Think about the condensation on the outside of a cold drink on a hot day... the same thing can happen to cold seeds, and that moisture would destroy them rapidly.

There are seed saving FAQ's (link in the top of the Seed Saving home page), but paging through them, I find very few references for saving vegetable seeds. There are, however, many instructive threads here, if you search for a particular type of plant. You might also want to buy or borrow the book "Seed to Seed", by Suzanne Ashworth, which is a great beginner's guide to vegetable seed saving.

The Seed Savers Exchange also has a wealth of seed saving info & tutorials on their website: Seed Savers Exchange

The only thing not mentioned is that many vegetables are sold immature (think zucchini, summer squash) and the seed won't be developed enough to grow a new plant. You may be ok with tomatoes, but most sold in the stores are hybrids so there's only a 25% chance you'll get exactly what it was (but you will most likely get an edible tomato). Heirlooms from Farmer's Markets are better if you want to save tomato seeds. Most peppers aren't ripe until red - so green peppers most likely will also have immature seed. This picture packed PDF applies to Peppers, Tomatoes and Eggplants. http://asi.ucdavis.edu/programs/sf/publications/pepperproject-savepepperseeds.pdf

If you're saving seeds that you purchased from a reputable company (mature and dry), refrigerate or freeze as Zeedman explained above, without exposing them to a lot of humidity or other source of moisture. Don't bother saving seed from hybrid or store-bought fruits -- the results are usually disappointing. If you're collecting seeds from your own garden: 1. Choose the fruits* from some of the earliest fruits, not the ones from later in the season that may not be completely ripe. 2. Mark the fruits you're wanting to save the seeds from with colored plastic surveyor's tape (inform your family that these fruits are NOT to be harvested for eating). 3. Allow the chosen fruits to mature completely -- this may be beyond the best-eating stage. 4. Harvest the fruits and collect the seeds (be sure to keep ID and markings with each group). 5. Process the seeds as directed elsewhere, and dry (maximum drying temp: 100F). Just for keeping for the next year or two, refrigeration is fine, but they do take up room if you have very many. 6. For the long term, use freezing; seeds should be about 14% moisture. A handy method to check for moisture is to hit some of the seeds with a hammer -- if they mash, flatten or just dent, they are not dry enough. If they shatter, they are dry enough to freeze. Seed that are not dry enough and are frozen will be damaged or killed. The optimum temp for freezing is 4F to 6F. A free-standing freezer usually stays colder than a freezer in a refrigerator. While many people freeze seeds in plastic bags, they aren't moisture barriers. Glass canning jars are ideal........................................................

* When I refer to 'fruit', I mean the seed-bearing lump of any type of plant, not just fruits like apples, berries, etc.

My wife was decluttering and found some tomato seed in some stuff she had taken from an old garage in 2013. The seed were in a pill box for some reason. She put some in a cup of soil and several have sprouted.