Knowing where the warning lights are can make at a glance scans for problems fast. High-end file servers are sometimes equipped with small LED readouts that indicate internal temperature, memory errors or other warning messages. Make sure you know how to manipulate the menus to get the vital information you need before communicating with technical support. Be sure to check around the back of the rack too. Network cards, power supplies and other components usually have lights that report on their status.
Drive arrays can report when a disk has failed and needs to be replaced.
Listen carefully for changes in the ambient noise in your data center. In a loud computer room it can be difficult to hear. One of my managers used to keep a stethoscope in his toolkit to check hard drives in massive drive cabinets. By holding the diaphragm of the instrument on each disk, he could verify if a drive was indeed spinning. Several years ago when modem pools were all the rage I used to be able to tell the negotiated baud rate just from the tones.