For most bill payments you can set up the payee right online, no need for even a phone call. The only time I need to go to an ATM is for cheque deposits or cash withdrawls. The only time I need to go to the bank branch directly is to see my bank manager about a loan or something. I have lots of rolled coins building up. This is all through the Royal Bank of Canada.
Similar services are available through all the major Canadian banks and trust companies.
I'm very happy with my experience. The jury is still out. I'm hesitant about paying bills online. You should make the leap.