There are a few different ways things can be done. However, the good news is that it looks like all whites are neutral, all blacks are always hot and the one red wire is switched hot. In addition, you only have the red wire on the receptacle but not black, so it is not a split receptacle. You need to do the following:
Receptacle - Change from switched hot (red) to hot (black)
Remove the red wire from the receptacle. It will not be used any more, so cap it with a wire nut and a piece of electrical tape to make sure the wire nut stays on, as wire nuts are not (usually) designed for a single wire.
Remove the two black wires from wire nut and connect them to the two hot side screws on the receptacle. Do not use the backstabs - not as reliable and not meant for multiple uses. Alternatively...
Add a short 12 AWG (15A or 20A circuit) or 14 AWG (15A circuit) piece of black wire to the wire nut connecting the two black wires (you may want to use a fresh wire nut) and connect the other end to one of the hot side screws on the receptacle. Then do the same thing for the white wires - remove from the backstabs, add a short white pigtail and connect all three with a wire nut and put the other end of the pigtail on one of the receptacle neutral side screws.
Switch - Connect to new cable
Disconnect the old red wire and connect the new red wire to the switch. Cap the old red wire with a wire nut.
Add the new white wire to the existing group of white wires.
It isn't clear exactly what is going on with the black wires. You have two existing black wires, and a wire nut but it doesn't look like a typical pigtail. It looks like one wire loops around the screw and then ends in the wire nut together with the other wire. What you want is the old two black wires + the new black wire all in a wire nut along with a pigtail which goes to the switch.
With a simple/"dumb" switch, there is no difference between hot and switched hot - i.e., you can flip the wires to alternate screws and it will make no difference whatsoever. However, with a dimmer, smart switch, timer, motion sensor, etc. there often is a difference. In your case, it is pretty straightforward: black = hot ("in", "line"), red = switched hot ("out", "load").
Many fancier switches (but not all) will also require neutral. That is easy in your situation (often it is not easy at all) - just add a pigtail to the neutral bunch. Watch out for wire nut limits - the package should list how many wires you can include in each type of wire nut.
Connect white to the neutral connection and connect red to the hot/switched hot connection. You won't use the black wire. This is based on the convention of "something other than black or white for switched hot, when practical." Note, however, that if someday you want to have a light and fan with separate switches, you can do that, using black for the light and red for the fan (or vice versa).
End result will be red wire between receptacle and switch boxes not used and black wire between switch and ceiling boxes not used.