Yes it will work with your home network. The only problem that you might encounter is that if the Wi-Fi connect is dropped and you are still browsing the internet, your phone would use the 3G network which would cost money, unless you are on a plan.
So to get rid of this problem there are a couple of ways to do it.
1. Before you use the browser, turn on "Airplane mode" which will disable all connections to the Verizon cell towers.
2. Block data through verizon. You can do this either in-store, online, or on the phone.
For more info on this use the this guide made by Fisharefriends, It is very useful.