Mlagma:
I then referenced the schematic for the Arduino Uno, and I saw that there are only two connections from the serial microchip in that schematic to the Arduino Uno (TXD and RXD). Am I right in assuming that the ATMega328P either does not have this capability or its unnecessary?
Well, partly right and partly wrong, but you are hopeless at reading circuit diagrams. 
Trace the connection from pin 13 of the ATMEGA8U2-MU , clearly labelled "DTR", through C5 and the confusingly drawn "RESET-EN" link to pin 1 on the ATmega8. This, plus "RN1D", a 10k resistor (as part of a "network" or SMD pack of four) is what you must replicate in your design.
Now, note that you do not need this connection for serial communication as such, because you usually arrange matters such that flow control - which is what RTS/ CTS is for - is not necessary, just as well because it is usually rather difficult to arrange correctly on both ends of the connection. You essentially do this by ensuring that the receiving device can handle as much data as you send it by buffering (which the "serial" library performs to a certain extent and your PC software does also), and that the sending device does not attempt to send more than the receiving device can handle until it acknowledges - if necessary - that it has processed all of the foregoing.
The connection from DTR to the reset has another purpose entirely - it is a mechanism to reset the chip to start the bootloader in order to receive a program ("sketch") download. If you do not wish to use a bootloader but will be programming your ATmegas exclusively via ISP (ICSP), or you do wish to use a bootloader on a very infrequent basis and are happy to manually use the reset button to do this, then you do not need that connection through the capacitor (but should nevertheless include the 10k resistor and possibly the rest button). Note that the reset can also be caused by simply shorting pins 5 and 6 of the ICSP header with a screwdriver.