To add another outlet what is the best course of action/can this outlet handle that many wires coming off of it? My thought is to pigtail all the whites, blacks, and grounds together and leave the red separate to maintain the switch. Is this correct?
That's probably your best bet. In fact the existing wiring with a split socket, and 3 blacks on the lower socket, is illegal. El cheapo 50 cent sockets only have 1 backstab per screw, and they've split the screws, so they're doing something dodgy as heck to get that third wire on there.
Better "spec grade" outlets provide a way to put 2 wires per screw without using unreliable backstabs, and that'll work for neutral, but they're going to have the same problem on the lower hot terminal. No option but to pigtail.
Now, you need to pay attention to box cubic inches because you have a whole ton of wires in there. I assume since they're backstabbed they are 14 AWG? OK let's count wires.
- Three /2 cables and one /3, so 9 wires.
- Grounds are "4 for the price of 1" so 1 wire.
- A receptacle takes a wire count of 2 wires.
- Cable clamps if they protrude into the box take 1 wire count for all.
I'm counting 13 "wires", and with #14 that is 2 cubic inches per wire so 26 cubic inches. That's a pretty big box.
I would search the box markings for its cubic inches (they vary on plastic boxes) and if necessary, add a box extension. The Legrand Wiremold Surface Conduit Starter Box is a tasteful one.