For mileage earning purposes, you'll probably be best with AA, so long as you do the following:

1) Be sure that as many of your flights as possible are sold with AA flight numbers (for example, HEL-ZRH, HNL-SYD or HKG-NRT, operated by AY, QF and CX respectively, but also marketed by AA under a codeshare agreement). You'll earn many more miles/points that way.

2) Sign up for a "platinum challenge". Provided you won't be spending too much time in Florida at the beginning of your trip, you should be enjoying elite benefits (such as lounge access) quite quickly.

Enjoy the trip...