Begin educating yourself about the BA Avios program. It's tricky, as it is decidedly different from other airline programs, but it does have some sweet spots. Wait for a promo and move some of your AMEX points to BA. Right now, you'll only get .8 Avios for 1 AMEX point, thus my caution to wait for a promo. You can then use Avios for AA and AS flights. Avios are charged per segment and per distance flown. I wouldn't want to use Avios for AA or BA redemption to Europe, though. The fuel surcharge will end up costing you almost as much as a coach ticket. You'll find better deals on coach seats domestically.

Should you want to go to Asia, consider the ANA program. You can move AMEX points to it. Do not do so until you are sure you can find your award seats. Once moved, any points in the ANA program expire in 3 years. It's been awhile since I've done an ANA booking, but I flew F to Singapore via Tokyo and then stopped in Tokyo on my return. I've also used an ANA award for a trip to the southern part of Japan, stopping in Tokyo on my return.