The 15-pin somewhat D shape port is VGA (sometimes called 15-pin RGB). That is analog video only. Some HDTVs have a VGA port for connecting a PC, but maybe the newer ones do not.
If the port with pins sticking out has 9 or 25 pins, that might be a serial port. Does your computer have a DVI port (sort of a D shape, but more squared at the corners). Maybe that is the long port you see (unless it is an old parallel printer port): https://en.wikipedia.org/wiki/Digital_Vi... The video signal on DVI is the same digital video signal used for HDMI, so if you have DVI on your computer, you could use a simple DVI to HDMI cable (which I am using) or adapter.
Normally DVI did not do audio, but some DVI devices can do audio through a DVI to HDMI cable. Otherwise you might need an audio cable (possibly 3.5 mm stereo plug splitting to red/white RCA plugs) unless you use your computer speakers. Since your computer does not have HDMI, if it does not have DVI, you would either need "vga to hdmi converter" (not a simple cable or adapter) or add a video card to your computer that has DVI or HDMI.
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