Before having kids I thought teaching them to clean up would be similar to the rest of parenting: once they're physically able to do it you start practicing with them, and after a while they're independent and do it reliably. You invest time and effort up front, but it pays back reasonably quickly with benefits for both you and the kid. While we've (n=3) had good success in some areas ( street safety, microwave usage, walking to school, tooth brushing, ...), tidying has not been one of these.
Early on I tried a lot of getting them to clean up, but it was very slow, tended to dissolve into battles, and didn't seem to be getting much better over time. Instead, we've mostly moved to finding specific places where they can take on a bounded responsibility. The goal is to give them practice without overwhelming them, and to use natural consequences to avoid fights:
Lily (11y) and Anna (9y) clear their places when they're done eating, carrying their plates etc to the kitchen sink. We started doing this after we got a pair of cats who love to get into human food, and it's become very important to get dishes into a state of low potential energy right away. If they don't do this there's often a crash and an urgent puddle of milk.
Lily had a sleepover last night. Her room is generally messy enough that there wouldn't be a place for her friend to sleep. But she understood (from past sleepovers) that part of hosting was cleaning to where there was space for an air mattress, and to where her friend would feel comfortable. She cleaned up without being asked, and was proud to show me.
Similarly, Anna wanted to turn on her AC tonight. She knew that a requirement for this was that she'd need to tidy a path between the door and the AC so I could come in and shut it off when I went to bed. She almost didn't turn on the AC because of not wanting to clean up, and talked with me about her dilemma, but did decide to clear a path.
Our dining room table is a place for both projects and meals. When we're getting close to dinner I'll often ask the kids "are any of these things yours?" and they'll put away their portion.
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After I tidy away large objects, I'll sweep everything remaining into a pile in the middle of the room; tidying a pile requires less willpower than repeatedly searching for things that need putting away:
Typically I'll then sort through it myself but other times I'll ask the kids to put away anything they want to keep from the pile, with the plan that I'll throw away the rest.
This does work, but the kids find it more stressful than I'd like, since they're worried I'll be throwing away something worth keeping. I've never actually thrown away something they want (since I keep asking "is everything left stuff I can throw away") but they still don't like it ("you wouldn't throw away that Duplo, would you!?"). Combined with often being in a hurry when I'm sweeping, I don't tend to do this much.
I'll often only be willing to play a game with them if they'll clean up their previous game first.
Sometimes I'll use the strategy of everyone cleaning up for a set amount of time (some people use a clean-up song, recorded or sung), but not often because it's hard to address shirking well.
Julia has a range of things she'll do with the kids, like sorting through their accumulated papers and projects. This post isn't an attempt at an exhaustive list.
Part of what makes all of this hard in common spaces is that responsibility among multiple kids is unclear. I usually can't tell who made what mess, but even if we agree on the facts it can be philosophically challenging. For example, say Lily gets out cloth, Anna starts playing with it too, Lily departs, and then Anna departs. Lily couldn't have cleaned it up when she was leaving without getting in Anna's way, but does Anna take on full responsibility for Lily's cloth fragments as soon as she starts to play? Should the kids race to be done first? Do you need to come back later and clean up whenever the other is finished? These are tricky questions where adults are generally able to navigate with some combination of designated responsibilities (ex: a specific person does dishes) and relaxed doing-your-part (ex: trying to clean more dirty dishes than you generate), but our kids are still on the young end for this.
The goal is still to get the kids to where they're responsible adults and good housemates, and I do see them moving in this direction. But I also think I'm being a bit lazy here, erring towards choices that minimize only short term work and conflict. I'm going to think more about how to be more intentional about growing their responsibilities here, cleaning up for them in fewer cases, and getting them closer to pulling their own weight.