The "relax" idea is easy to say, harder to do. Very important. Baby can pick up your vibe.
Longer term: The #1 best thing to give your child is to have a strong relationship with your partner. IMO, this takes a lot of work, especially if you were raised in a home without that. As much as education, safety, and health, this is the highest gift from parent to child...the capacity to have healthy relationships.
Second only to that is to earn respect from the child. Enforce boundaries and consequences ...lovingly. Learn how to fashion consequences to fit the crime.
Lastly, if you let your child learn how to divide and conquer parents, you are asking for mountains of conflict.