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.)
gardening.stackexchange.com/users/51/bstpierre
In theory yes.
In practice, often no.
A lot of annual vegetable seed that you get are hybrids, which means that they are not genetically stable.
Next years plants will not be as productive as this years, if they are productive at all.
This is especially true of conventional seed, which the seed companies intentionally breed to under perform in future years so you have to keep buying seed.
It is less true of heirloom plants.
So if you have heirlooms you're more likely to have success with seed saving.
gardening.stackexchange.com/users/124/Daniel Bingham
Yes you can!
I haven't tried storing tomato seeds myself, but I have tried to store pepper seeds (and they are closely related).
This hasn't worked for me, but I suspect in most cases, the seed was immature.
I do, however get "volunteer" (i.e. accidental) tomatoes and peppers.
These are typically fruit which have come off and left on the bed (or lightly buried) or they've survived in the compost heap.
Tomato seems to do better from the compost heap, whilst I've found peppers seem to do better just being left or lightly buried.
This year I've had over half a dozen paprika peppers grow like this
- I assume that is the variety, I haven't got any fruit yet.
If they are paprikas, they are second generation from a volunteer paprika pepper grown last year!
I've also had store bought Fresno peppers grown in the compost.
Not sure why those particularly - a lot of pepper seed goes in the compost, both store bought and home grown.
(store-bought pumpkins also produce seeds which survive the compost heap)
The one vegetable where I've had a lot of success with storage, are black beans.
I just put the end of the crop in a paper envelope (as beans without husks), and hang them on the noticeboard.
They dry out naturally.
One year I mistakenly put them in a sealable plastic bag and they just went moldy.
Unlike peppers and tomatoes, I usually get very good germination rates with stored black beans.
gardening.stackexchange.com/users/69/winwaed
I consider seed collecting to be a part of the whole experience of gardening.
Experiment a bit.
If something works, keep collecting those seeds, if another one doesn't, go on and try harvesting from a different plant.
I think it's just more rewarding when you get a plant from seeds you harvested and sowed yourself, the same way it's more rewarding to pick that first tomato plant from the vine rather than from the produce aisle.
Also, it give you something to do as your garden is winding down for the year.
gardening.stackexchange.com/users/93/rsgoheen
I think it is still feasible to store seeds until the next year.
This is provided that you have the correct storage measures taken.
They must be stored away properly in the right container like an airtight and moisture-free jar and away from damp/humid areas in the house.
Avoid putting them in the fridge as the temperature in there might spoil or lower the quality of the seeds.
gardening.stackexchange.com/users/2134/ThomasMaloney
This discussion of seed storage  got me wondering if it is feasible to harvest seeds from vegetables (I'm partial to tomatoes) as I cut them up for consumption.   Generally, I buy seeds each year. But if I have a steady supply, and it's not ill-advised, could I not save seeds for the next year? If so, what steps should I take? Is there a reason  not  to save vegetable seeds?
Can I harvest vegetable seeds and store for the next year?
gardening.stackexchange.com
<vegetables><seeds><seed-saving>
gardening.stackexchange.com/questions/767
gardening.stackexchange.com/users/101/kmm
Saving seeds for next year is not difficult as long as you aren't using hybrid varieties. Some plants are hard to get seeds from if you live in a cold climate or require special techniques. Seed collecting is part of the whole experience of gardening.  It is so much more rewarding when you get a plant from seeds you harvested and sowed yourself. I think it is still feasible to store seeds until the next year. Saving seeds for many vegetables is not difficult, but you need to start with open pollinated varieties and not hybrid varieties. They must be stored away properly in the right container like an airtight and moisture-free jar and away from damp/humid areas in the house. Some vegetables are difficult to save seed from, especially if you live in a cold climate and for some vegetables, there are special techniques to harvesting seed. For some vegetables, there are special techniques to harvesting seed. A lot of annual vegetable seed that you get are hybrids, which means that they are not genetically stable.  
I think it is still feasible to store seeds until the next year. Saving seeds for many vegetables is not difficult, but you need to start with open pollinated varieties and not hybrid varieties. Some vegetables are difficult to save seed from, especially if you live in a cold climate and for some vegetables, there are special techniques to harvesting seed. S16: For some vegetables, there are special techniques to harvesting seed. A lot of annual vegetable seed that you get are hybrids, which means that they are not genetically stable.   Seed collecting is part of the whole experience of gardening.  It is so much more rewarding when you get a plant from seeds you harvested and sowed yourself. They must be stored away properly in the right container like an airtight and moisture-free jar and away from damp/humid areas in the house.