Backup says it was never backed up - Immediately after a backup

Hello,

My computer is getting older so I wanted to get better about frequently backing up my files. I recently bought a cheap WD My Passport external hard drive and I set it up with the backup assistant to backup my entire computer every day. Recently I've grown a bit concerned though due to the following... After the computer runs its backup which it does everyday it gives me a small error saying "Check your backup results" when you click it it shows two files that were not backed up (two pictures on my desktop which I cant even find even with show hidden files enabled) so I'm not sure where its even finding those. Regardless, I assumed it was backing up everything else and just not backing up those two missing files which is fine. Following the backup it shows the hard drive is about half full, so I assumed that meant it successfully backed up most of my information. However, in the back up and restore menu it says "Last backup" and it says "Never" Despite the fact the computer keeps running through the backup and continues to add stuff onto the external presumably due to the decreasing in space after each update.

I'm just nervous if my computer dies down the road I will go to retrieve information from the hard drive and it will be empty or not work. Any idea how to ensure this backup is successful?

NOTE: I am a complete newbie when it comes to windows. I primarily use a Mac (insert mac joke here).
So please take it easy on me from a technical standpoint.

Thanks!

Honestly 90% of the computer I could care less about. There is only one folder on my computer I am most concerned over having backed up - the folder titled "Thesis". That being said, I have been considering doing exactly as you just mentioned - I just wanted to check to make sure there wasn't a setting or something I was doing wrong that could be easily fixed.