I need to find out the service tag of my Dell server or computer. How do I find out my nique five- to seven- digit alphanumeric (letter and number) service tag code without looking on the back of the system or opening door of bezel?

You need to use the dmidecode command. It is a tool for dumping a computer’s DMI (some say SMBIOS) table contents in a human-readable format. This tool can display serial number or dell service tag without looking on the back of the server. Type the command as follows as root user:
# dmidecode | grep --color -i serial
OR
# dmidecode -t 1
Sample outputs:

# dmidecode 2.9
SMBIOS 2.6 present.

Handle 0x0100, DMI type 1, 27 bytes
System Information
	Manufacturer: Dell Inc.
	Product Name: Precision M6500                 
	Version: Not Specified
	Serial Number: 12AB69D
	UUID: 44454A4C-3100-109C-1032-XXXXXXXXXX
	Wake-up Type: Power Switch
	SKU Number: Not Specified
	Family:  Precision Workstation

You can use Dell service tag # 12AB69D for downloading drivers or for other support related issues on http://dell.com/ itself.