By the primary grades, families can release more responsibility to children. These include helping them learn how to prepare (and clean-up) their own snacks and helping with family meals: setting table, carrying cold dishes to the table, helping clear the table and helping load the dishwasher. Those minutes helping in the kitchen also offer an opportunity for 1:1 conversation about school, their sports, etc.
Other household jobs include making their bed and keeping their rooms picked up. True, they won't resemble Martha Stewart's organization, but, a few rumpled covers or clothes piled up rather then strewn around the room are a beginning to children taking responsibility for their space. You can increase your expectations from there.
As for school related tasks, expect your kids to load and unload their school backpacks, hand over notes from the school, unpack homework and remember to carry library books and picture money to school. True, you may find a couple of green sandwiches in the bottom of those backpacks, but, hey, a science project can be started with them.
When you rescue your kids, you aren't developing their skills with responsibility. Yes, they may not be able to get a 'new book' during library class, but the day they do return the book, there may be time to check out a new book. (And, as far as picture day money goes, in most instances you can call the school to make arrangements for paying late.).
So, start relinquishing your responbilility to develop theirs. Before you rescue ask yourself, 'is this helping my child become independent?'