Kerri,

Your flying OneWorld, SkyTeam and Taca who has a reciprocal miles agreement with Iberia (OneWorld, like BA) and United (Star Alliance).

If you want to spread the miles, sign up for BA for the LHR-LAX routes, and your travel on LAN. For the AeroMexico routes signup with Air France-KLM (Flying Blue). As I am guessing your LHR based these two make the most sense for your locations.

For your flight home from Rio, book with BA, the flight makes a one stop in San Paulo , then non-stop to LHR.

How do you book this flight? Contact the financier of your travel.