I have a 3 year old too, I try to get him involved with stuff as much as possible so that household things can be done together, so he's with me getting attention as my OH is also out long days, but things get done...! of course this depends on your child! I do work full time and can't afford cleaner and have no childcare outside of working, so if I'm honest I do almost all cleaning with him with me so obviously you don't need to do as much with yours there but my ideas below
Daily stuff -
After eating I will get him to help me by putting plate in the sink then I stand him on a chair while I'm next to him and we wash up together, if he's taking a while eating sometimes I have already done a chunk of it but he can do his bits and then help me dry up.
He has a mini hoover (a Henry) and he will hoover with that while I hoover with the big one, he lifts the mats up etc to hoover under and I show him little bits of dirt I know the toy one can pick up so he gets into it
He splits the clothes into dark/light/white and puts them in the machine, I let him put the liquid etc in with me helping and do the settings on the machine too and turn it on
He's also the 'peg master' and brings the box of pegs out and hands me them telling me their colours while they go on the line, or in winter just loads up the drier lol
He helps bring the clothes and he puts underwear and pjs away in drawers (messily) while I hang stuff
He also has his own broom and dustpan and brush
When he's in the bath I do stuff like bleach the toilet quickly and wipe round the sink etc
Non daily stuff -
As he likes water I play to this and have a mini spray bottle with some water and fairy in and get him to spray the back door/kitchen cabinets and wipe them while I clean the higher stuff properly.. I only do this in the non carpeted rooms as it does get a bit wet lol
Changing bedding - I get him to try to get the pillows out of the cases and into new ones (I mean it never goes very well but it keeps him busy) while I'm doing the rest
He has a toy lawnmower to mimic while it's being done properly, as well as letting him loose with the watering can while I just sit and watch
I'm crazy blessed with a child who seems to like getting involved with this stuff though, but he's also always had to watch me do stuff round the house (and dad on weekends) as the only time we have is while we are with him, so I think we've always just given him little jobs while we do it and now he tends to help with everything..
In terms of day planning - on the weekend when I'm off work but my OH isn't we normally get up, have brekkie, do a few bits together (and he sometimes gets bored and plays toys, I don't massively ask him to help it's just always on offer if he wants), then talk about where we are going and make the house tidy with toys in boxes before we go out, I don't go out generally with toys on the floor and we have canvas boxes with lids on so he can easily just throw them in so we can go to XYZ
I think keeping on top of the little things helps and trying to make it less of a chore :)
Oh - cooking I missed, my other half works every other weekend, so if he has him I bulk cook some fresh food for freezing (curries/bolognaise etc) or if he's got weekday off he does. They are used some days; other days things that can go in the oven are done so they cook while doing other things - e.g. I'll put a whole XL chicken in with some potato's, and use leftover the next day with just cut salad (sometimes badly cut if I have a helper), or with a jar of curry cause and some microwave rice. And when I don't have time to do that, then freezer fish fingers and chips cooking in the background won't kill anyone ;)
See what you may find that works for your life/routine, I have to say I just do stuff at whatever time whenever depending what I/he feels like, but if structure would suit you more then maybe list things down and highlight those you think you can do with little one, and those you can't, then see what little one will do with you/and ones where it makes it way harder to do together and move any that can't be done to the days when they are at nursery.
When they are at nursery my number one suggestion would be shove music on and do the stuff that needs doing at mega speed with loads of effort at the start of the day, it builds up a sweat, you get it done fast, and then you feel rewarded by getting you time after, if you put it off until the end it's torture!