you are accessing the internet via wi-fi because you have wi-fi turned on; if you want to access the internet via Verizon's cellular network, you need to turn that feature on (depending on the phone you have, it's under Settings, Data Usage ... ensure that Mobile Data is on) and turn off wi-fi
You posted this in the Android section, so I'm assuming you have an Android smartphone? If so, then you do have Verizon cellular data ... a data plan is required to activate any smartphone.
I have a Droid 3 and cannot access web info when I am connected to my wi-fi at home. My router and modem are fine.
I can only connect to the internet when I turn off my wi-fi and go directly through Verizon 3g? I have tried to reset the phone and it will no let me update any system changes? Any advice/help would be greatly appreciated.