The server and network security problems you should understand are file integrity, privilege escalation, applications not loading, network file/share access, the inability to open files, excessive access, and excessive memory use. The causes of common security problems include active services, inactive accounts, anti-malware configuration, misconfigured permissions, open ports, and rogue processes.
A system administrator has a toolbox of tools that should include port scanners, sniffers, a Telnet client, and the application of cipher tools and checksums.