As a mom of three boys I know that toys seem to multiply over night and can easily take over a home. Parents understand that kids are going to have toys around, but we don't want it to get so out of control that tidying up is almost impossible. Getting the toy situation under control really isn't as hard as you may think.
Depending on the age of your kids, you may or may not want to have them around. In my opinion, really young children should not be involved with toy purging. They will want to keep everything. The older the child the more they will be able to make choices about what they really want and like. Only you as the parent can determine how much involvement you want to give your children.
Start with a box for trash, a box for giveaway, and a box for "lets think about it". I guarantee as you're going through the toys you will find rocks, trash, broken pieces, and junky KinderEgg and McDonalds toys. Don't waste one second thinking if you should keep that stuff. Getting rid of those things alone will make a big impact. Then you want to separate the toys into categories like: stuffed animals, cars and planes, Legos, puzzles, action figures, animal figures, etc. Once you have your category piles, decide which items really matter. Do you have multiples of the same thing? Does it work? Do you have all the pieces? Does my child actually play with this? Remember: kids play better with LESS stuff around them. If you have a gazillion little pieces scattered around, they will be overwhelmed and not know where to start or how to play.
Now put the toys you want to keep into bins by category. Large, general toy boxes are NOT the way to go unless its only for large items like stuffed animals. Little Timmy will never find that piece he needs if everything gets dumped into a big box. By using separate bins for each category, it makes clean up a lot easier. For really young children use clear bins or put a picture of what kind of toy goes it each bin.
A tip about stuffed animals: they take up a lot of space. And they collect so quickly because everyone wants to give a cute stuffed animal to a kid. Pick three or four of their absolute favorites to keep on their bed. Give away as many as you can, but with the others (if you really can't let them go yet) get a big cloth bag with a drawstring or an empty bean bag chair. Fill it with the stuffed animals and your kids can use it to lean against or lay on. At the very least they will be contained in a bag instead of spread all over the floor.
For LEGO I suggest shallow bins (or multiple) for storage. If its too tall and deep it makes it harder to find pieces.
Attention sentimental parents!! In my experience I have found that parents often have a harder time letting go of toys than kids do. We remember the birthday or Christmas when our child got that stuffed animal or how they used to play all the time with that toy set. Its hard to let go of those younger years. I suggest choosing one bin to keep for sentimental items and LIMIT YOURSELF TO THAT BIN. Keep baby's first teddy bear and their favorite book or whatever but let go of the rest. Do you want the space or do you want to hold on to the past that is already gone? Your child probably doesn't have the same attachment to those things, so its really for you, not them. Think of the end of Toy Story 3 and let those toys live their best life with another happy child! :)