Clear out stuff that's broken or not used - give away, sell or store.
Invest in proper storage - shelves with baskets work for us (IKEA), doesn't have to be expensive. Make sure DCs can reach it all to get toys out and put back themselves.
Use (and label with words/pics) separate drawers for different toys, i.e. one for playmobil, one for bricks, one for train set, etc. This makes the toys much more fun to play with as they're not all jumbled up, and also much easier to tidy up. Involve DCs in deciding what goes where, get them to help make labels for each drawer, etc. That way it becomes their system too.
And finally, insist on tidying up every night. It won't be easy at first, but stick to your guns and they will get into the habit eventually. You will have to help them/oversee the process, otherwise they'll get bored/distracted. Make it fun/competitive - 'I bet I can tidy up ALL the dolls before you can get all the lego off the floor', or sing a tidy up song together (we use Frere Jacques - tidy up now, tidy up now, tidy UP, etc - it's got a hypnotic effect, honest!).
Second what PotPourri says about them playing better with tidy/fewer toys. The process of getting there is a P in the A to start with, but once you get into the habit it's so much better for everyone.