Things to look: Miss-configured services (cronjobs), incorrect file permissions (exportfs, sudo), miss-configured environment ($PATH), binary with SUID bit, software or OS with known vulnerabilities.
First try simple sudo:
$ sudo su -
What can we run with sudo?
$ sudo -l
Try su as all users and the username as password
What services are running as root?:
$ ps aux | grep root
Look for vulnerable/privileged components such as: mysql, sudo, udev, python
If /etc/exports if writable, you can add an NFS entry or change and existing entry adding the no_root_squash flag to a root directory, put a binary with SUID bit on, and get root.
If there is a cronjob that runs as run but it has incorrect file permissions, you can change it to run your SUID binary and get a shell.
The following command will list processes running by root, permissions and NFS exports.
Use netstat to find other machines connected
Command to skip ignored lines in config files
If Mysql is running as root, you can run commands using sys_exec(). For instance, to add user to sudoers: