First of all, my hat is off to you for doing it all without a dishwasher! I'm a SAHM mom of twin boys (18 mos.) and I don't think I could do it without the dishwasher.
I'm sure others have made great suggestions, so in the hopes of not being too repetitive, here's what we do:
1. Crockpot/bulk cooking - we make a lot of things like stews, gumbos, etc. that have meat, veggies and a grain cooked right in. It saves time, money and dishes. Sunday is cooking day for us. We keep some in the fridge for during the week and freeze the rest so it doesn't go bad. We like spicy, so we cook things to the boys comfort level and spice it up for ourselves. One of our favorite cookbooks is One Pot.
2. Bins - we sort toys in clear plastic bins. They can be stacked and put in a closet or left out. The kids can see what is in them and it's easier to put things back at the end of the day. We've started making it part of our bedtime ritual to put up the day's toys. Frankly, the boys are too young to be much of a help, but they get a kick out of it.
3. Wet wipes - I bow before the person who invented those things! We get the Kirkland brand at Costco (sooo cheap) and I use them to clean stovetop splatters, the front of the fridge, dropped items on the floor. They really help tide me over between house cleanings. Also, the boys love to "help" me clean. I give them wet wipes and ask them to clean off low tables and windowsills. They're not half bad!
4. Laundry - ugh. I NEVER liked doing laundry, even before kids. When I lived in NYC, I found a service that cost the same as a laundromat and managed to go years without touching a washing machine. Sigh. Those days are over. Nobody warned me that the laundry would increase the bigger the boys got. Well, my trick is to stain treat everything while dad gets the boys in their PJs. When their hamper is full, I wash it all in cold. No sorting! Then dad and I fold while the kids are either asleep or playing together and put it away asap. For my husband and me, we are each responsible for our own laundry.
5. Paper - the amount of the stuff that comes into the house is terrifying! We have a bin and a shredder. The moment mail, notices from preschool, etc. comes into the house it either goes in the bin (trash), in the shredder (promo mail), in the magazine rack, or on my desk (action items, like bills, school notices).