To enable Ethernet access on older Macs like these, you must plug in the cable first, then restart so that it sees an ACTIVE connection at Startup. Then it will allow you to use the TCP/IP Control Panel to set the Interface to Ethernet. When you have a Router, the easiest way to configure is "Using DHCP". You must close the TCP/IP Control Panel and Save Changes for the changes to take effect.
Now do some action the requires Ethernet Access, such as setting Network Time or opening a Browser. If you get on the Internet, all is well.
If not, open the TCP/IP Control panel and see what IP address you have been assigned. 169.254.xxx.yyy indicates you are talking only to yourself. You likely still have physical problems such as bad cabling, wrong kind of cables, bad port on your Router, or crossover cable used where straight or patch cable is expected.
If you have been assigned an address that is in the range of your Router, often 192.168.0.xxx and a subnet mask that matches the top three bytes 255.255.255.0, you should have your Internet connection, but you may not have a Domain Name Server (DNS). Entering:
http://17.149.160.49
... should get you Apple, Inc. If it does, you need to enter your ISP's Domain Name Server (DNS) info into your Router so that it is dispensed with every IP address.