No, wifi is completely seperate from the cellular network. It will not cost you anything, but will depend on whether or not there are wireless routers/networks near you and if they are secured (with a password) or not. Wi-Fi is basically a home/office internet connection, just wireless.

For example, all starbucks has 'wi-fi'.

If you don't have a data plan for your phone, get a data block immediately from Bell. They dont like giving data blocks, and they say some functions wont work but they mean stuff that connects to the internet, which will cost you if you wifi isn't connected since it defaults to the cellular providers internet. It cost be about $12 for less than half a MB of usage. Their rate per MB is more than $50, im not kidding.. just checked my bill. If you just got the phone you can get the charges reversed just say you didn't know it would connect to the cellular web and that it cost so much etc etc. They should be lenient if you just got the phone etc...