I assume you want to use The Philippine's official postal service.
I looked up all their post offices in Manila and this was the closes to Paco:
Municipality : 2F Robinson's Otis, UN Avenue, Manila
Address : Robinson's Otis
The cost of a postcard to send to the USA is:
up to 20 grams: 24.00 ₱
21-50 grams: 58.00 ₱
51-100 grams: 80.00 ₱
101-250 grams: 147....