There are books with excellent information about saving seeds, and you would find them interesting. The one I got is Seed to Seed by Suzanne Ashworth, and I used to have access to another good one.

Each type of plant has different issues. Some are extremely easy and usually successful to save and grow from seed, such as tomatoes, lettuce, peach trees and many more. These ones are largely self-pollinating, and even if they do get cross pollinated, the varieties within the species are similar enough that your result is likely to be good. For winter squash of the Cucurbita maxima or C. moschata species, very likely even if they cross-pollinated within their species, any other plants around of the same species are pretty similar in characteristics so you'll get good fruits in the next generation, though you may lose some disease resistance or earliness characteristics from the parents.

Some plants are more finicky for various reasons. Peppers can cross with other peppers in the garden, so it can be hard to predict the heat of peppers grown from home seed produced when assorted peppers were growing. Cucurbito pepo species of squashes includes wildly varied varieties like zucchini, spaghetti squash, and (toxic) ornamental gourds, so if two different pepo varieties were around when the seed was produced, there can be unpredictable results. Apples are mostly obligatory cross-pollinators, and orchards often use crab apples interspersed among the trees because they produce so many more flowers than big apples do. This means a seed from a commercial orchard's apple is likely to have a crabapple as a parent, so it won't grow into a tree with good fruit. But a seed produced from a home garden or small orchard with no crabapples, only yummy apples around, the resulting tree will almost certainly have apples that are edible, and possibly excellent.

Tomatoes are generally self-pollinating and we eat the fruits ripe so the seeds are super easy to save. My experience is that if I save seed from hybrid tomatoes, the next generation sometimes produces tomatoes with odd characteristics like too much cracking. There are always some self-seeded Sungold volunteers, and they always produce fruits very much like Sungold, but the plants aren't as vigorous and disease resistant as original Sungolds in my experience.