How many pins does the "serial" port have? A serial port usually has either 25 or 9 pins.
If it is sort of a D shape and has 15 pins it is likely a VGA port. If that is the case you would need an HDMI to VGA "converter" that would convert HDMI digital video to VGA analog video. That cannot be done with a simple cable or adapters since the video signal is different.
If that is the only port the monitor has, it may not have a DVI port, which is a bit more rectangular and has more pins, or at least more places for pins, since it can potentially do analog and digital video. If the port is DVI, you would only need an HDMI to DVI cable, since the digital video on DVI and HDMI is the same.