If the seeds are from hybrid veggies, don't waste your time. If you want to save seed, go with heirloom varieties.

Open pollinated peppers and tomatoes should be good.

Squash are trickier. You have to make sure it doesn't cross with another squash if you want it to breed true. This can be done with careful work. Tear open an unopened male flower, collect pollen on a q-tip or small brush, tear open an unopened female flower, pollinate it, and then wrap the female in a loose bag (paper or cloth) to prevent insects from further pollinating it with other pollen.

Corn will work if isolated well enough, but will work better if you bag it until silks are ready to accept pollen, then pollinate and re-bag.

I had multiple spaghetti squash come up in the compost last year....2011.....

Didn't know what they were, but waited, and delicious!!

The seeds could only have come from store bought spaghetti squash as I had never grown it.

If your plants are not hybrid, yes. That is, if they were labeled heirloom, open-pollinated (OP), something like that. You may still have some diversity in leaf shape etc. but it is even interesting.
If the original seed stated it was a hybrid of any sort, or maybe "F1", then it is simply unpredictable. May still be nice, I did get some very curious bicolor tomatoes and big pumpkins that way. Peppers may be sweet with Very Hot seed, for example.

I have kept mature dried up Okra pods and used the seeds year after year. These were clemson spineless. Okra prefers a hot environment with tons of sun though.

I like to save Cilantro seeds (aka 'coriander')and resow at three week intervals, also Parsley, Fava beans, tomatillos, Roma tomato, pumpkin and others. We've had carrot go to seed and actually got excellent volunteers from outside of the raised beds...beets did it too and we always thinned both so meticulously for years, but the volunteers just grew before we knew or made any effort, so we just harvested, and harvested and harvested and they did great, tasted great etc...

Many of the greens (Kale, chard, collards etc.) are very easy to save seed from. But, these tend to hybridize easily so plant one type per year or growing season so that they don't flower at the same time.

Here is a link that might be useful: Edible Garden Seed Saving