Yep, big topic. Some things you can propagate from seed, some you can use tubers or rhizomes or bulbs. Some may do both.
It can be really easy and obvious - for example, last year I let several of my radishes just go ahead and live out their little plant lives. They grew amazingly large greenery, flowered and set seeds in cute little pods. I let the plants die off, cut them off at the ground and hung them to dry for some time, until I got around to picking the pods and shelling the seeds from them. I had started with one store bought packet of seeds, I collected at least twice what had been in the packet. I planted those seeds this spring and they are doing great.
Squash would be another pretty darn easy one, or melons. Let the fruit ripen on the vine. When you eat it, wash off the seeds and set them aside to dry. Easy to locate, easy to identify, etc.
Onion is a bit more of an issue. They will flower and set seed, but you have to be willing (like with the radishes) to forego harvesting some of your onions and let them go to seed.
Again, last year I had lettuce that bolted. I let it go ahead and set seed and just spread it around the bed, in hopes of getting some this year. Jury still out on that
But, really, while it is a potentially complex and involved process, and some plants (cucurbits!) are notorious for cross-breeding all over the place, it is also just as simple as can be.
Experiment a bit and see what you get
And as you said, you can save money, but you can also get a
landrace variety of plant that is well suited for your specific conditions.