I think there are two aspects to keeping the house looking good - the cleaning and the tidying. They really are different. If you can declutter, it makes the tidying that much easier. Which, in turn, makes the cleaning that much easier. I am not a particularly tidy person and, like yours, the kids leave a trail of mess behind them so I have a couple of things we do (or try to do, and need to do more) that help.
A few years ago we got one of those footstools that has storage. LIFE CHANGING. We shove all the extra little blanekts etc in there and so we just have to do a quick plump of cushions to get the lounge tidy.
We also have deep drawers in the lounge - where we can dump stuff and then I try to go through them every now and again and ditch what's not needed. It's on my list to do as the last two times it was DH who did it and he finds it hard to throw things away so I feel like the clutter is b uilding up again! 
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Ditto, school bags etc just get chucked into the cupboard under the stairs.
Robot vacuum. Game changer. I often spend 5 minutes in the morning making sure the floors are clear then turn it on while I do school run/shower etc. Makes a huge difference. Our stairs aren't carpeted so I'll often just quickly brush all the dust/dirt from them onto the floor in the hallway which then gets vacuumed up by the robot. I should mop more often but it gets down once every two weeks and sometimes every week. The robot does do a really good job of keeping it in good shape though.
Bathroms - like others, I am regularly just sticking toilet cleaner down to keep things clean. A couple of times a week I do a quick clean of the the seat (both sides) using a bathroom cleaning spray and toilet paper and I usually do the sink at the same time. It genuinely only takes 5 minutes or less. I also regularly spritz the bath before I get in the shower then just sort of rub and clean it with a cloth using my feet while I'm washing my hair. A proper clean of the full bathroom happens just every two weeks by doing this.
If you aren't already, get into the habit of properly cleaning the kitchen after dinner every night. Wipe down all services, stove, wipe down of any obvious spills or ssticky cupboards etc. This means that "deep" cleaning wher eyou're moving all appliances around and the like really doesn't need to be done more than once every fortnight or two and by doing it routinely at the end of every day, there's no really caked on grime to sort out.
Having said all that, I DO find it can start to get out of hand quite quickly if you're not doing all these things really regularly. And I AM lucky enough to have a cleaner who comes in every fortnight and does those deep clean things as otherwise I would say that would be at least one weekend morning a fortnight I'd be doing it.