How to save/move photos on a mac?

I’m sure that macs explode during the summer season for graduation gifts / college fall semester. So heres my n00b question.

I got a mac a week ago, and I’m having problems saving photos to an album, or transferring a photo from the “transfer” photo album to a legitimate album. Or after I transfer the photo from the “transfer” album, I can’t delete the original photo without losing the one in the second, correct album.

Can I get some help? How do I save my photos to an album, and not have to transfer them. Also, can I delete the original photo and keep the duplicate? Thanks