My parents had six children and we all have at least four years of college. They weren't wealthy, but they were very good to us. What they did, and what I'd have done had I had children, was as follows:
1) We got a savings account as soon as we got a social security card. Gifts from relatives all went into this account from day one.
2) When we got old enough to understand the concept of money (and wanted to spend it), half of anything we got (e.g. allowance, gifts, lemonade stand profits, or pay from our jobs) went into the savings account. They'd eventually start making me make the deposits myself.
3) When time for college rolled around, we were responsible for paying for one year and they covered the rest. They helped with fixed expenses (they paid for a car and insurance, and provided room and board while I was on campus), and I worked summers and part time during the year to pay for books, gas, eating out, and eventually my room and board when I moved off campus.