I saved spinach seed last year for the first time, and yep, it was easy. It took longer than I'd expected, considering how eagerly spinach can bolt sometimes. The plants were ultimately quite big, so it needs more space than when you're planting just for eating. Of
course you can plant closer together, thin and eat, and leave some spaced out to go to seed. The seeds are very spiky (I think I heard somewhere that smooth leaved spinach makes spiky seeds and savoyed tends to make smoother seeds). From a little patch of about 1x3 or 2x3 feet, I got a large cup of seeds, which will last me for several years if I don't give it all away. A lot also fell around and self seeded, which was fine.
To clean the seeds, this worked well for me. I dumped the rough seeds and chaff in a big sieve and stirred and pressed it with a stone, mortar and pestle style. Most of the chaff broke up small and fell through, and I picked stems off the top, and the ended up with fairly clean seed.