If you want to pursue status with AA, I would immediately obtain an AAdvantage credit card, and start charging on it to earn Loyalty Points. I don't know whether the Citi and Barclays versions are available to residents of Puerto Rico, but there's always the Banco Popular version as an option.

If you can get status with Hyatt through your travels, that might trigger a cost-free status match/challenge from AA, given the close AA-Hyatt partnership.

If you have a couple of trans-Atlantic roundtrips in Business Class coming up soon, you could pretty quickly achieve oneworld Sapphire status by crediting to Iberia Plus. But then you'd be acquiring Avios which, depending on your redemption patterns, might not be as valuable to you as AAdvantage miles. On the other hand, OWS status earned through a foreign FFP would get you lounge access when flying AA domestically.

Good luck!