My kids are 3, 3, and 5 1/2. I think they do a lot more than other kids we know in real life. I don't expect them to do any of these jobs without support and reminders. I do, however, expect them to do the jobs without raising a fuss.
For us, I think it happens easily because it's just the way things always have been. I start them on putting clothes in the hamper, as soon as they can crawl and pull to stand. Then I introduce the other jobs regularly. If a child refuses to do the job, or raises a stink about having to do it, I deal with the situation on a case-by-case basis. Sometimes I let it go, if my judgment tells me that's the right thing to do. Sometimes I decide to stand my ground and insist, in which case I withhold all other non-essential activities until the job is done. (By non-essential I mean I don't withhold meals or sleep. I wouldn't do that. But I will staunchly refuse to allow any other toys to come out, the TV to be turned on, a child to go outside to play, etc. until the job is done.) For me, having them be responsible for a share of the work is important enough that I'm willing to go to bat to make it happen.)
However, lest you start to think it's a constant power struggle, let me tell you that this "go to bat" scenario has only happened three times in 5 1/2 years-- twice with DD1 when she was 3, and once with DS when he was 2 1/2. Most of the time, I just have to let them see the consequences of not doing it-- once I woke DD1 up an hour early in the morning to show her how her kitten was yowling over an empty water bowl, and often I'll just let the toys accumulate a few days until the child complains about them, then gently point out that they could be cleaned up easily.
DD1:
changes her bed when she wets it
can run the washer and dryer, and is expected to wash her own bedding with help
puts her own folded laundry away (I sort and fold)
clears her place at the table, and takes one other item to the kitchen
helps set the table on request
cleans up her own toys
keeps her own side of the bedroom clean (she shares with the twins) with reminders and help
can vacuum under her chair if she spills food
puts her own dirty clothes in the hamper, right-side out and with her socks bundled into a pair
feeds her kitten and gives her water every evening
hang up her own towel after a bath
put things like wrappers or used cups in the trash or sink, rather than leaving them wherever they fall
The twins:
clear their own place at the table and take one other item to the kitchen after a meal
help set the table
clean up their own toys, with assistance, before naptime and bedtime
help by running errands in the house-- like "please go in the basement and bring up a jar of tomatoes" kind of stuff
put their own dirty clothes in the hamper
hang up their own towels after a bath
put things in the trash, or in the sink, when they're finished with them rather than leaving them wherever they wind up
I don't offer tangible rewards for household chores. I do offer verbal validation-- "It makes me really happy to see all the toys cleaned up," or "When people help it makes my jobs so much easier." Those are nothing but the truth.