Some research into this on your part would probably be helpful, but I'll summarize the situation for you.

British Airways will charge "fuel surcharges" on certain long-haul flights using their planes or their miles (avios). Typically, you can expect to pay about $500 per person per leg from a city in the Continental US to any city in Europe if the transatlantic part of that trip is on a BA plane, even if you're paying with AA miles. To avoid this, what you want to do is ensure that your flights are all on AA planes, or at the very least, make sure the flight over the Atlantic is on an AA plane, even if you connect in London on a BA plane to your final destination.

To quickly search for these optimal flights, do an advanced search from the AA.com main menu, select to use miles, select a date to start with (just guess something close). At the bottom of the page, in the Airline box, select American Airlines instead of All Airlines. Then click search. You'll get the same calendar search you normally get, except it will only show date availability when there is at least one option where all flights on are AA metal.

If you want to fly somewhere that AA doesn't fly to directly, but BA has flights to from London, first do a search from the US to LHR, ensure THAT flight is an AA flight, and then find a separate connection from LHR to your desired destination. Once you've found a date where both flights work, go back and repeat your search but select all airlines, and on the specific date you chose, find the connecting flight option that has the AA on the transatlantic leg. There will still be a cash component of the charge, but it will be substantially less. Coming back will be somewhat more expensive unless you can avoid British airspace on your flight back, but still less than a BA flight.

Hope that all makes sense.