On The Iphone 3g Is It Free To Use Wireless Internet

I am thinking of buying the Iphone 3G (refurb) for only 50 bucks since its discounted when i renew my 2 year contract. At my house downstairs we have a Modem, and the upstairs computer has wireless internet. Would i get to go online on the Iphone for free at my house, as well as Wi-Fi hotspots?