If you're new to programming, I suggest writing your algorithm out in psuedocode first. Your requirements might look like this:

open INFILE for reading
open RESULTS for writing
foreach machine_name (in INFILE) {    # loop over lines in input file
    results = LoggedOnUsers(machine_name)
    print results to RESULTS 
}
close INFILE
close RESULTS
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That's the basis of your algorithm (without any error-checking — you'd want to add this). I'd look at the following for the syntax:

perldoc -f open
perldoc perlsyn          # For the "foreach" loop
perldoc Win32::NetAdmin
perldoc -f print
perldoc perlintro        # May be of use
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I'm pretty sure that it doesn't matter which machine you run this from. I'm also pretty sure that you'd have to run this as a Domain Admin (to have rights to the LoggedOnUsers function)
Welcome to the world of scholastic monastics, btw.