You currently have two cables in the box. One is the power source, and has just a black (hot), a white (neutral) and a bare (ground) wire. The second cable has a black (hot), a red (hot), a white (neutral) and a bare (ground).
As currently wired, the incoming black is attached to the outgoing black and to the switch via a pigtail. This makes the unswitched half of the outlet always live and provides a hot connection to the switch. The red wire is attached to the other side of the switch and provides a switched hot to the second part of the outlet. The neutrals are connected together in the box and the white wire at the outlet itself covers both halves of the switch (only the hot side has separate connections).
Adding other switched lines is fairly straightforward (this assumes that you have sufficient amperage to cover the additional fan and light).
As you surmised, you need a three wire (plus ground) cable. The new white is connected to the existing whites. If you pick a triple switches with a common hot terminal (most are configured this way), just move the existing hot pigtail from the old switch to the common terminal on the new switch. This provides hot to all three switch units. Then attach the red wire for the outlet to one of the other switch terminals, and the new red (from the fan or light) to another switch terminal and the new black (light or fan) to the last terminal. You can swap these three wires to assign whichever switch unit you want to each device.
Connect all ground wires together.
You are good to go.
[I will try to add a diagram later if I have time.]