logger logs to syslog facilities. If you want the message to go to a particular file you have to modify the syslog configuration accordingly. You could add a line like this:
local7.* -/var/log/mycustomlog
and restart syslog. Then you can log like this:
logger -p local7.info "information message"
logger -p local7.err "error message"
and the messages will appear in the desired logfile with the correct log level.
Without making changes to the syslog configuration you could use logger like this:
logger -s "foo bar" >> /var/log/mycustomlog
That would instruct logger to print the message to STDERR as well (in addition to logging it to syslog), so you could redirect STDERR to a file. However, it would be utterly pointless, because the message is already logged via syslog anyway (with the default priority user.notice).