Company leave is 2 weeks at vacation pay with certain blackout dates. On top of that there is Gov. Paternity leave which varies quite a bit from state to state so look into which base you plan to be at (I know NY and CA have the longest leaves but I'd advise against commuting just for a leave).

Beyond this though look closely at eligibility. I know the company one only applies after you've completed a full year on property. States vary but you need to have worked for a certain period before it kicks in.

In my own experience and the experiences of guys I talk to AA is incredible about giving you time off for family stuff. I wouldn't worry about needing to take off time Beyond what you are explicitly entitled to, just work with the flight office. That being said it will be unpaid so once again look into your eligibility and budget accordingly.