Actually, I don't think you can operate a "successful" (with considerations for the long-term development of the child) star chart. Since you asked for "better ideas," I propose:
- connecting with your child by setting aside one-on-one time
- being kind (read connected, respectful, etc) and firm (known as authoritative parenting style, not to be confused with authoritarian)
non-evaluative statements that encourage the child to reflect on their own behavior instead of somebody else's evaluation of them. This looks like:
"I notice you washed the car this weekend. Thank you."
"I notice you put your toys away."
"I notice you didn't really eat much at dinner"
encourage your child using the following:
"I appreciate _ "
"Thank you for _ "
"I have you seen do _ before so I know you can _ "
"I have faith in you"
As for why reward charts, praise, etc don't work, I recommend reading Carol Dweck's research as well as Brene Brown. Both of them just released books that are very accessible to the average reader. Basically the problem with praise/rewards is that (and most of these apply to punishments too, which is why I use neither):
It invites the child to wonder what is so wrong with them that adults around them have decided they need personal cheerleaders?
It invites the child to choose between the reward and their sense of dignity, putting them between a rock and hard place, holding a double-edged sword.
It encourages the child to become a puppet/people pleaser and invites perfectionism and "praise-junky" behavior that leaves the child desperate for others' approval
It discourages the child from trying new things as learning is fraught with mistakes and disappointments and the child hasn't developed the coping skills for this, and may become afraid of disappointing others and not getting praise. This is what Carol Dweck calls a "fixed mindset."
It can lead the child to confuse between "I did good" and "I am good" which maybe doesn't sound that bad until you consider the flip side of "I am bad." This is what Brene Brown defines shame as and it can have debilitating effects lasting into adulthood. Her work parallels nicely with Dweck's.
And, lastly, it doesn't actually teach any skills so at best is a waste of time. Rewarding "good" behavior, isn't the same as teaching conflict resolution, self-regulatory, emotional literacy etc skills and this can show up later in the classroom environment especially.
So, bottom line, sticker charts and their ilk may seem "successful" in the short term, but in the long-term have unintended consequences and may cause quite a bit of harm.
Instead, I recommend running your household the same way you would an ideal office environment: with dignity, respect, shared vision, transparency, communication, consistency, structure, trust, compassion, etc.
And, none of this means you can't celebrate genuine accomplishments but these are not planned beforehand and instead arise naturally.
All that being said, each of these: "sustained reduction in fighting, consistently less nagging required at getting dressed time, regular help with laying the table etc" warrants a separate question and should be dealt with by teaching the required skills. A sticker chart does not teach conflict resolution skills which require complex behaviors, attitudes and skills, for example. I'd be happy to answer each of these questions directly.