You can buy hybrid seeds anywhere in large quantities and the plants all turn out about the same, but when you save seeds, they all turn out different. How do the professionals save the seeds? If I could do it I would save a lot of money.
Essentially, professionals keep "parent" plants that will produce "child" plants that are genetically similar. Hybrid seeds are seeds from the parents that produce those child plants. Every time they need more hybrid seeds, they have to induce and wait for the parents to flower and fruit.
Hybrids are generally assumed to be "unstable", which means that they will not produce seeds that make plants just like them. Good seed shops will specify explicitly that something is a hybrid, and if that's mentioned in the variety's description, you can assume that it will not breed "true".
Other varieties have been bred continuously within the same genetic pool for long enough to be true-breeding; many heirloom varieties fall within this category. Their seeds will generally produce very similar plants generation to generation.
For tomatoes and peppers, at least (and many other plants), if you have the ability to keep them indoors through winter, you can keep cuttings of them pretty much indefinitely, if you are careful about keeping them disease and pest free. That would let you keep a strain of plant that you like without having to continually buy more seeds.
It depends. It may not make sense for you to try (too much work, or too many plants needed for good seed).
A hybrid is a cross of two different lines, and may well not breed true.
For a discussion on the broader issues of saving seed, have a look at Gardening When It Counts, the author talks about saving seed, he ran his own seed company for a number of years, so he does have a good bit of experience. He talks about how many plants you need, hybrids, etc. It is a complex subject, it may be that you cannot, or that you can for just one generation (plant the bought seeds, collect seeds from them, use those until the are gone over the next few years).
Here is his website, I've not looked at it before today, I don't know how useful it will be to you: Soil And Health Library
Good luck, and let us know how it works out.