If your home Wi-Fi is saying your phone is is connected but no internet, then you need to check your landline internet connection. Either you need to reboot your modem (cable or DSL or other service). It might be a good idea to reboot the router at the same time if they are two separate pieces of equipment.
FYI this wasn't a wireless cellular problem. It was a problem with your connection to the internet via the landline.
I'm most definitely NOT a VZW employee. If a post answered your question, please mark it as the answer.