As a family of five it’s not always easy to work out the best/cheapest way of getting from A to B. Sometimes a shuttle bus is cheaper than a taxi (if you have to upgrade to a 7 seater). Local transport like buses and trains are usually the cheapest, but not always the most practical. Do a little research on TripAdvisor forums or ask the hotel or Airbnb host you have booked with if they have any suggestions. They are usually a good source when it comes to local transport and airport transfers.
If you want to cover ground quickly hire a driver/guide. This was by far the best choice we made when travelling around Sri Lanka. We had 2.5 weeks and would have spent a lot of time on trains/buses. Instead we crammed in ten destinations and multiple activities, all from the comfort of an air-conditioned van, with free wifi and a wonderful guide. If we had a month to do the same trip, the experience of travelling as the locals do probably would have been our first option.
Here are some popular sites to help you get around.
Train travel: