Yes, you need a BlackBerry data plan to use the internet. The cellular network handles phone calls and text messages only. Everything else is data.
Most phones get their data from their carrier. With BlackBerrys you get the data from RIM through your carrier. The good news is that you don't need a separate smartphone data plan, just the BlackBerry plan.
Normally you need a BlackBerry data plan to use WiFi. The data you use over WiFi won't cost you anything and won't count against your monthly data allotment, but you need the BlackBerry data plan on your account to get access.