My kids are older elementary--can't remember when we started, but probably once both were in elementary school. We split allowance three ways: 1 part is savings, 1 part is charity giving, 2 parts spending. We give low amounts but the kids still have more than they need, and one of my kids often chooses to put more of the spending money into savings. They decide how to allocate gift money.
Savings: Intended to be longer term, but no goal at this point. I remember my siblings and I saved up and bought a keyboard when we were in middle/high school--it wasn't something my parents were going to buy us, but they had no problem with us having one. I see my kids' savings the same way. I'm happy to have them save up for something that is important to them. We've said that they don't need to save for college, in part because that's so a huge goal and would be overwhelming for younger kids. If necessary, I'd only introduce the need for college savings in high school, when they could actually earn bigger money towards it.
As the kids' savings have grown, we've helped them to invest a small amount in a pretty safe investment so that they can get a taste of how the stock market works.