Ugrading hard rive in mac mini

Ugrading hard rive in mac mini

I recently bought a new mac mini (my first mac ever yay) with a 1TB hard
drive at 5,400 rpm. the guy at microcenter also sold me a 1TB hard drive
at 7,200 rpm and said it shouldn't be to hard to replace. He said first
I'd need to back up the OS to a usb thumb drive and then physically
replace the hard drive and then some how get the OS onto the new hard
drive.
So far I've found a few tutorials online on how to physically replace the
hard drive but it seems like a lot more work than I anticipated (ie.
http://www.ifixit.com/Guide/Installing+Mac+Mini+Late+2012+Hard+Drive+Replacement/11716/1).
Although, it does seem like something I could do.
However, my main difficulty now is i'm having a very hard time finding any
help online on how to 'back the OS up on a usb thumb drive' and how I
should re-install it once I do put in the new hard drive. He mentioned
using time machine but when I open time machine it doesn't give me any
options to interact with the USB thumb drive.
My last part of the question is that I purchased apple care for 3 years
and I just want to make sure that none of this will somehow violate the
warranty.