There are lots of ways to get miles. If AA is the airline you choose to go with, check out the AAdvantage section of AA.com, which lists most of the main ways to earn miles.
But, I should point out that getting award tickets to Europe during the summer is VERY difficult. As you point out, AA miles can be redeemed/earned on partner airlines (i.e. British Airways to Europe). And getting 5 award tickets on the same flight may prove to be a real challenge.