Although I'm not a 'beginner' I still have lots to learn about bitcoin.
1. How would I go about downloading the necessary parts of bitaddress.org to generate an offline paper wallet?
https://github.com/pointbiz/bitaddress.org2. How can I ensure that the download is safe?
The github source is code reviewed and monitored. As long as the checksum matches, you can be relatively confident that you didn't get malicious code from a man-in-the-middle attack.
3. Where should I download the file(s) to? (Would a USB suffice?)
Depends on how paranoid you are. Do you have any concerns that the USB will have malware that will autoexecute when you plug it into the computer? In general a USB should be fine.