In DS1 there was a penalty to your current covenant if you left it for a different covenant, but it was harder to join a covenant. In DS3, it is much easier to switch via equipping. If I start ranking in a covenant and receive a reward, then switch to a different covenant, will I loose rank or the reward from the first?
No, there isn't a penalty. Additionally, you can turn in covenant items to gain experience in covenants that you are not currently in.
EDIT: It seems a little relevant to include that increasing your rank in some covenants can lead to negative consequences with some NPCs.
Sirris really does not like if you level up Rosaria's Fingers