From LHR, I would prefer to connect in JFK, as connecting in MIA can be very painful. Bags will be tagged to the final destination, but as any other US connection, you still have to clear immigration, claim your luggage, clear customs, and re-check.

If this is your first time to the DR, and you prefer a quiet place to enjoy with the family, you should look into the southeast region, specifically La Romana (Casa de Campo) and Punta Cana (Cap Cana: Golden Bear Lodge/Caleton Villas).

If you are flying to La Romana, your options are either connecting in MIA or flying to Santo Domingo (SDQ) or Punta Cana (PUJ) and having your transfer from the airport to La Romana arranged by the hotel.