Shallow baskets with open tops. They can be lined with fabric so little fingers don't get piched.
Also you can take sturdy storage boxes - like those photo boxes and cut a "window" that you cover with - well, sorry, it's a little plastic - plexiglass or heavy mil clear plastic tablecloth (you can get that at the fabric store). Then you can see what's in them but it's not all plastic. Or you could glue on a picture of the contents.
Drawstring bags will work for some things. Dolls can go in those corner "hammocks" you can hang in the corners of the room.
You can also sew pockets on a big piece of fleece or flannel and put toys in the pockets. I once dreamed that if my kids ever had a big enough room, I wanted to decorate with beadboard wainscoting and use a strip of denim all the way around the room as the "border" but I wanted the denim border to have pockets all the way around so it would also be functional! (It never happened!)
www.target.com has some canvas bins, click on the storage section, and they should be in there. I have also seen them at Bed Bath and Beyond. They seem great fro organizing, but I am lacking bookcases.
2 Ikea Expedit bookcases. They are open cubbies, some have baskets. The smaller expedit has space for 16 cubes (not all would be reachable by a young child); the bigger size has more cubes (25?). They aren't cheap, but for the amount of storage space they give, they are waaaay less money than similar systems from places like Pottery Barn.