Pretty much any smartphone can connect to WiFi, which is a wireless networking technology commonly used by computers. Your phone can connect directly to a wireless router in your home (no need to go through your computer), and there are also public wireless networks in businesses (such as hotels, coffee shops, restaurants, book stores, and more), churches, libraries, universities, etc. There are also WiFi networks which require a password, but friends, family, etc. might let you use their network when you come over if you ask. Note that if you use someone else's WiFi network (such as work or school), your activities might be recorded and monitored.
To read about how to use WiFi on your particular phone, see pages 176-178 of your user manual. If WiFi is completely new to you, even on computers, this introduction might help. Also, JiWire makes a good app for finding nearby places with public WiFi networks.
It's possible that you already have a wireless router at home and don't know it. The modem from your ISP (Internet service provider) may or may not be a wireless router as well. Or you may have something that says Linksys, Netgear, D-Link, Belkin, 2Wire, or Apple AirPort on it, or some other Internet-related thing with antennas sticking out of it (but not all wireless routers have external antennas). Look on the sides, back, or bottom for the default network name (SSID) and password (wireless network key).