using the Atmega328P
No, you don't need to upload the Bootloader, it's already in the chip.
BUT, when you unplug the ATMega328 you no longer have the serial connection support over USB. 

If you upload a sketch while the ATMega328 is on the Arduino board and then put it in a breadboard, it will work providing you don't need serial/USB and you have the correct oscillator (16Mhz) on your breadboard. 

There are a couple of approaches to create your own breadboard projects:


Keep your Arduino intact and buy some ATMega328 chips. You can use the Arduino as an ISP to program the chips without needing a serial/USB connection.
Use an Arduino Nano. This plugs directly into your breadboard and has everything you need onboard.


Start reading here, this has all been done before.