On your actual photo, the weird looking green port (topmost left side) is a serial port.
Following downward, the next two ports are Display Ports - Those are monitor hookups. You only need use one (probably either, though sometimes single monitor must go on DP port 1 [Probably topmost]).
If your existing monitor has HDMI - Either an inbuilt cable or a port - will determine whether you need an HDMI to DPM (DP, DPMI) adapter (cheap, 10 bucks or so) or an HDMI to DisplayPort (DP/DPM/DPMI [all same thing]) cable (probably under 20 bucks), or some combination thereof, keeping in mind M/F connectors...
If your existing monitor does not have HDMI, it may have VGA port (small, blue) or a DVI port (Larger, white) - It is possible to go from either one to DisplayPort with a proper adapter, though VGA would not be recommended. DVI will do for jump starting you, but even that is old tech, and you need to start scrounging for a newer monitor, capable of HDMI or DP.
Did you glaze over yet, or did you pick up what I'm putting down?

I mean, it's me... I start talking geek and go right over folks' heads.
Standing by...