The first part of this answer is easy: enough to keep the kids occupied so you can read in peace, or take a shower alone, or just hear yourself think for five damn minutes without somebody yelling "mom!" or crying or fighting or bleeding or stuffing socks down the toilet - but not enough toys to make a mess. Right?

The second part of this answer is also easy: you need quiet toys, building toys, creative toys, thinking toys, musical toys, and imagination toys because this is what you want your kids to grow up learning. And books. But don't forget that back in the day kids had way, way, way fewer toys and were thus forced/encouraged to use their imagination and creativity to find things to do instead of relying on cheap plastic crap made in China and enriching the Chinese government so that one day they can Take Over The World. Oh my. Where did that come from? 

Now comes the hard part: how many, where do they go, and who puts them away? Easiest first:

Q: Who puts the toys away?
A: Why, whoever uses them of course, because...of course. I cross my heart promise you that if you start and enforce this one simple habit for the first five years of your child's life, you will save yourself years of pain and suffering. Years. A decade or two. Even longer if your kids God forbid end up in your basement after college. 

Teach your children to put things away as soon as they can hold an object in their hand and walk, crawl, or drag themselves across the floor to the shelf/box/bag/tub/whatever. Make it a game. Join in the "fun" every time at first, and then again as needed.

Be consistent.

This will work.

However — and it's a big however — accept the fact that one day oh so soon the "fun" of clean-up time will just plumb wear out because...it just does. Eventually you'll most likely need to add a consequence for those times the darlings don't feel like putting things away. The secret to peaceful clean-up times is to have your consequence thought out in advance and to Stay Calm. Calm. Very calm. Very matter-of-fact: if this, then that, it's just the way it is. For example: If you don't want to put your toy away, then you won't be able to play with it next time so I'm putting it way up here. Then, next time they ask for it, remind them that it's out of reach because they didn't want to put it away and that maybe tomorrow they can play with it again if they put their toys away today. Calm. It's not a punishment, it's a consequence, a choice your child made. Don't get angry, don't raise your voice, don't threaten or beg or cajole. And if they never ask for that toy again? Well, cool. One less thing to take up space and clean up after. 

Q: Where do the toys go? 
A: In the easiest, closest, tidiest places possible. If you expect a small child to stack up the blocks neatly down the hall in the closet, rather than tossing them in a bin right there, you'll be eating dinner at midnight, if ever. 

Depending on the layout of your home, you may have one play area or several. Young kids want to be near mom and dad, meaning you'll want a few items in whatever rooms you're typically in. While it would be oh so lovely to have matching bins and cubbies and shelves, it's not necessary. Every year or so most of the toys will change anyway as kids move up in skill level, meaning you'll constantly be figuring out what bin, what size, what shelf will fit what toy. The only "rule" is to keep like with like; which teaches your kids basic organizational skills. Use a low basket on the floor for crawlers. Low shelves or bins for toddlers. They need to be able to see the toys, reach them, and put them away all by themselves.

Q: How many toys do you need? 
A: Fewer than you probably think. For toddlers and preschoolers with a designated playroom, maybe start with five "activity" items in that room. Choose things that have a longer interest factor: Legos, bricks, blocks, puzzles, cars and trucks, dollhouses, barns, coloring, and other items that require creativity, encourage making things up, and can expand and grow. Put a couple books in their bedroom; one or two (not ten) items in the tub; big cardboard bricks in the laundry area; a few low cupboards in the kitchen so they can empty and rearrange while you're working. Rotate other toys in and out as their interests change. Pay attention to what your child really gets into and start by feeding that. See what they ignore or toss aside and pull these items out from the rotation. Always, always store some items away so you can bring them out for days of desperation, car trips, restaurants, quiet time. 

So what do you do if you read this post too late and you already own 3,246 toys, ​