I've had several comments all relating to storage issues. I'll address each of them in separate posts. Kids clutter solved here!
First things first, EDIT your belongings and those of your kids. We all live with too much stuff and I know that when I'm faced with a storage problem the first thing I do is pair down what I have.
Go through your kids toys (when they aren't around) and put everything they don't love or play with on a regular basis into one of two boxes. The first box is for things he/she has outgrown and won't play with again. You can put those away for younger or future kids, or donate them to someone who will play with them. Be fierce in this editing. Keep items that are sentimental in a box (in storage, attic, etc.) for each kid (really for you!).
The second box is for things you aren't sure about and he/she may miss them if taken away. Keep that box of stuff, but hide it and see if your child asks for "X" item. After a few weeks go through the box, keep what you love and rotate it out with other toys every once in a while. That will refresh his/her toys on a regular basis. Donate or give the other items to a friend.
Many of our families give us way too much stuff for our kids. It comes from a good place, but aside from finding room to store it, it can overwhelming to kids and can teach them that they "need" all of this stuff. My kids play with the same few items -- many of them simple, everyday things -- day in and day out.
Okay, you've edited and now you need storage for your living space. My first preference is to use adult furniture to store kids toys in a living room. I have three tall bookcases that have drawers on the lower half and shelves on top. I use the drawers to store the kids toys. I put their books on my cookbook shelves in the kitchen (where they can reach them). That keeps our living space kid-friendly, but adult-pretty.
If you don't have any such options in your living room try one of these, depending on your decor, leather ottomans that open for storage, wicker baskets with or without lids, metal bins (found at Home Depot, very affordable and modern looking), or baskets that fit in shelves you already have. Consider putting large pieces that can't go in a drawer in a bedroom.
As far as placement of these containers, try to arrange your furniture so that you can place a few behind a sofa or bookcase -- using the sofa to create a "wall" for your storage space. This won't work if your sofa is against the wall, because you want space for your child to access the toys and put them away.
Also consider what toys can go to a grandparent's house if they live close by. I will look for some pictures of storage items I've described and post those soon.
Hope this helps a little. Edit, sort, donate and find adult-friendly storage. Good luck.