You need a display, a keyboard, and a pointing device (mouse or trackpad). You may want to also have an external DVD player and a printer (wired or wireless). Specs for the mini are here along with a picture of the back so you can see the ports.
In terms of hooking up the display u have 2 options:
one is to use the HDMI out port from the mini.
the other is to use the display port/Thunderbolt out port.
I add the following as it might help.
I started last week with my new rig by using the HDMI out port and it has the effect of disabling any sound control via the mini, its all through the display.
The Apple remote wouldnt work for sound control.
I wanted to use the mini for face time and not having a usb headset I bought a usb sound card for 9 euro and switched the display from the hdmi to the display port.
It now all works from the keyboard/remote/headset