You can report to someone (often your direct boss), and you can hear a report of some event or other.
A report can be about or (more usually and more formally) on something.
You can write a report for some client publication, employer or interested body.
On your specific question on the 'on' vs 'of', the most usual use of 'of' is with the noun form of 'report': "several reports have been coming in of disturbances on Clare Street." If you want to express a similar a similar meaning using the verb form of 'report', treat it as transitive verb ("the police report disturbances on Clare Street") or use a subordinate clause ("the researchers have reported that the new antibiotic is effective against Streptococcus resistant to penicilin.")
To report on an event often means to provide further information about an already identified event:
"The police had brought the violence under control by 2:30 this morning; Christina Aguirre is on the ground to report on the aftermath. ... Over to you, Christina."
To tell someone that a report deals with a particular subject, just use 'on':
"The research team have just published a report on the effectiveness of the new antibiotic."
I should mention that 'of' can also be a synonym for 'by' or 'written by':
"The second report of the Commission on Safety in the Workplace."