Logs for processes, firewalls, IPC programs, logins, routers, WWW, mail, file transfers. Keep a Sys Admin log for everything you change on the system. Keep those handy scripts running for disk usage and scan the filesystem for file permission problems. Create new user accounts, directories, permissions on files, delete and remove accounts, reactivate accounts that have been disabled.
Monitor processes running - remove stale or zombie processes.
Filesystem free space state, keep an eye on quotas Monitor file permissions of users, no 777 or 666, watch out for SUID and GUID programs. Keep current on current security expoloits for the services that you run. Make sure those services are running and performing properly. Field E-mail questions and problems from users, management, faculty. Understand the need, create a plan and implement it.