Getting admin privileges on a mac with almost no access
There are three avenues:


get the firmware password or bypass that 
socially engineer the person or team that locked you out (try convincing them you have a decent need and can be trusted before you consider trying to blatantly deceive them)
hope the os or third party software is unpatched and there is a known privilege escalation vulnerability you can exploit. (Assuming you can’t get the NSA or similar to help you)


In short, there is no easy way since you’re up against Apple engineering and trying to defeat the security of the system. 

Bypassing firmware: https://support.apple.com/en-us/HT203409