1) Check that you have a valid network interface that's recognized by the system (obtainable either by popping open the network manager used by your distro or, the lowest common denominator, open a terminal and issue /sbin/ifconfig -a). A wired connection should show up as something akin to ethN, where N is a number (usually 0-based, so the first device is usually eth0). Wireless is a bit more varied and depends on the wireless card you have, it can be wlanN, athN[/n], ethN, usbN, and perhaps a few others.
2) Check that the connection you have from 1) has a valid IP address (either a static that you've been assigned or, much more likely, an automatically-provided one from DHCP). Again, either use the network manager to gather this or use /sbin/ifconfig $INTERFACE
3) If you have a valid IP, try pinging another machine on your network (for example, if you have a router that is doling out IP addresses, try to get to the router's configuration webUI)
4) If you can get to the router, try pinging a known, external entity that will respond to pings. I always just issue the command ping www.google.com
5) If you can ping google (or someone else outside of your network), then you should be golden, and realistically if you're still having issues we'd need to pop open some weapons-grade tools like wireshark to sniff out the issue