External drive for specific files only - not for whole back up (Time Machine)

Hi. I use my personal MacBook Air for some projects at work. I want to start using an external drive for storage of those files only (I've been using flash drives, but it's out of control).

I want to specify which files to move store on the external drive. Because it is my personal computer, I do NOT want the whole computer backed up on this drive. But every time I connect to an external drive at home, the back up starts. If I choose not to back up, I can't use the external drive.

Thanks. I'm not good at this stuff (obviously).