- show running-config: displays running configuration form RAM on the router.
- show interfaces: displays information about the router interfaces, including encapsulation, address configuration and whether the interface is up or down. I use this one a lot!
- show arp: displays any address resolution protocol entries learned by the router.
- show ip route: displays routes manually configured or dynamically discovered by the router.
- show users: displays any users connected to the router.
- show version: displays the version of Cisco IOS software running on the router, name of image and amount of RAM.
- ipconfig (ifconfig on Linux).
- ping to test Layer 3 connectivity and basic DNS functionality.
- tracert (on windows)
- netstat show information about devices communicating with a host, including IP address and TCP port information.
After all this you can test Layer 7 connectivity by accessing the router through SSH, Telnet and a web browser.
There's also some debugging with the debug commands, for example debug ip rip. Stop debugging with undebug all.