In the context of someone's appearance, these words are all positive, but they do each have a distinct meaning:
Good-looking: pleasing to the eye, attractive.
Beautiful: not merely pleasing, but delightful visually. It's stronger than "good-looking". We describe uncommonly attractive or very good-looking people as beautiful. It is usually reserved for women. "Handsome" is preferred for men.
Well-groomed: Used to describe someone who pays attention to their personal hygiene. There is no bad breath nor any unpleasant body odor, their hair is cut and styled and any facial hair is controlled. It implies control over and taming of nature.
To look put-together: This means that you look tidy and that your outfit is coordinated. Everything fits well together. It doesn't look messy, sloppy or as though you've put no thought into what you are wearing. The clothes are usually clean, ironed, and your general appearance is well-groomed.
Chic: elegant, well-dressed, though not necessarily fashionable.