It appears that credit cards will be a major source of miles for you, and that American Airlines will be your best choice of airlines, especially if you want to use those miles for a nice international trip. You have received very good advice on credit cards: Starwood for American Express, and a Visa or Mastercard for those many merchants who do not accept American Express. The obvious choice for the latter is the American Airlines Mastercard. Now, be sure you get the maximum bonuses for obtaining those cards: 10,000 points for first use from Starwood, and 20,000 miles for charging $250 on the American Airlines card. Links to both offers can be found in the Credit Card section of my website below. Starwood is on the Free Cards page, and American is on the Annual Fee cards page (though both are free for a year).

After you receive those cards, consider applying for the other cards of Starwood and American: The Starwood Business card (another 10,000 points), The American Airlines Business Mastercard (another 20,000 miles), and the American Airlines American Express card (another 15,000 miles). I say "after you receive those cards" because your credit score may cause banks to balk at issuing you too many credit cards. Nail down the two important cards you are going to use, first.