I am on my third toddler.... what has helped is minimise any toys that have the potential to be a huge mess/hassle/get damaged easily. Have a few of these, but only ones that are actually fun and not a huge pain to set up/put away. They need to actually earn their keep. Also, keep these high up so they can't be got out, everything that is accessible is tough and easily slung back into boxes.
Easy access boxes/tubs/drawers, especiallly if labelled with a category is helpful. And toys like stacking rings, duplo, bits that click together etc should be stored together as much as possible, if you have a set of stacking rings and all the rings are distributed between 5 different boxes, they can't be played with and the rings are just annoying clutter in the other boxes. Stack them back together and put up out of reach then bring down for a game.
We tend to just clean up in the evening, but if we need space for a specific game then I might go in and clear things to make a space for that - this is another benefit of having the fiddly things up out of reach as it means you can say let's do marbles/a jigsaw/this board game - but we'll just tidy up first. It shows them there is a point to tidying. I don't do loads of persuading them to join in/threats etc as have found this doesn't work, instead I just do it myself (modelling) and explain why - they tend to join in anyway this way once they are used to it being a normal thing. Also once about 2-3+ if you explain that it's quicker when we do it together they are even more motivated to join in.
Avoid complicated sets with loads of different pieces that only work in that set - some of the "character" big pieces like Paw Patrol/Peppa Pig/whatever are fiddly and have stupid bits that fall off or get broken - I don't buy them as a rule.
Consider toy rotation - I don't do this but it works for a lot of people. What I do is declutter fairly regularly and get rid of the outgrown/broken/junk toys (like McDonald's freebies - some get kept if they are actually played with) plus don't go overboard at Christmas/birthdays. Keeping the overall amount of toys to a manageable level, so even if every single thing is upended (it happens) it doesn't take long to put right is key.
Clear up immediately after meals so that food does not make its way to toys and drinks are less likely to be spilled - I suck at this, but it helps a lot when I do it. We got a dustpan and brush with plastic bristles that is ONLY used under the highchair and that helps. Also, we had one of those plastic mats for wheely office chairs, put that under the highchair to make a wipe clean floor. It's still disgusting, but it's easier to clean - I never got on with the shower curtain method.
DH tends to clean up the front room after we put the kids to bed but recently he's been doing it before bed and involving them - he is good at summoning energy and enthusiasm to make it a game/challenge whereas I am not at that hour.
My thing is to try and keep on top of the kitchen/dishes, and washing. I do it around them - generally tend to give them something to play with in the room. DS3 is a tricky age for doing the dishwasher as he wants to grab all the knives so that is harder now and I tend to stack on the counter and DH will fill it when he gets home. Then DS1 empties it.