Kids are messy. No way around that. But as parents we do have a tendency to clean up after them because it's faster and more practical. Just like anything though, kids need to learn to clean, they cannot invent skills just like they can't learn to read on their own (well, most of them anyway).
There are different ways to go about it. A lot of people use rewards combined with chore charts. While I respect this method, I don't completely believe in it for our house as I think my kids need to have a sense of belonging to our family and its responsibilities.
There are ways to teach kids to clean that don't necessarily have to include rewards (or punishment for that matter). Of course, just as with anything, consistency is important and that's what eventually sinks in. And if you haven't started this routine from the start, it might take a while for them to adjust.
• Model what you want to see your children do. I always clean up after myself wherever I am. I make my bed when they are up and can see me do it. I also usually do it while singing or telling funny jokes so they get a sense of it being relaxed and part of life, not so much a burden.
• Organize toys. Children need structure in which they can feel comfortable. If there are clear places where to put different kinds of toys, your kids will more easily and more quickly put them away. Everything has its place.
• Include them. Most of the kids I know love to help out. Especially with cleaning. I let them spray and wipe surfaces, vacuum, clean dishes, sort out and fold laundry etc. And I let them do it with loud music when it's daytime and preferably during the weekend. They feel responsible and are proud of it. Putting away their own clothes that they folded in their own drawers makes them feel really proud of themselves. Granted, you can't be too specific about how the drawers are organized, but it's theirs, so ultimately they're the ones picking their clothing; I don't have to see it!
• Routine. So that I don't have to tell them what to do, and therefore not give them the sense that they are being nagged, we democratically decide who does what, but it is always done in the same order. For example, we always undress the table the same way (like most people actually). First the plates get emptied in the trash, then they go in the dishwasher (or sink to get washed later), then we put away the food, and finally we clean the table. The order is always the same like a choreography, and the only thing they have to do is choose who does what. It's a dance.
These are some things working well in my home, what about you? Do you have a system that works for your kids?