You're using data regardless - websites are made of files sitting on a server, and those files are made of collection of bytes - whenever you use the internet on ANY internet-capable device you are using data. PC, laptop, PS3, Xbox, BlackBerry, anything - all use data when they connect. The transport is the question here; your phone should default to WiFi if it's enabled, you have a profile set up, and in range of the WiFi network. You don't need to turn off the mobile network, it should do it automatically. The reason WIFi doesn't count against your carrier's data limit is because your carrier doesn't have to deliver data to you if you're on WiFi, whatever Internet Service Provider is providing your WiFi network delivers it.
When I'm mobile, I'm on Verizon's 3G network, they are my data provider and all my data usage goes through them. Once I get home my device connects to my WiFi and my data usage is provided by Cox Communications. Does this help clear things up?
Posted from my CrackBerry at wapforums.crackberry.com