I forgot to answer all of your questions. Getting on ship early, I can hardly restrain myself, sure you can usually board earlier than the posted time. Avoid the queues, get a drink, explore the ship and the Diamond is a big ship! Take in the views from the top deck and get ready for the sail away.
Organised tours just really depends on where you are going and how experienced you are. The advantage of the organised tours is that they guarantee to get you back on time. In less developed countries relying on old vehicles to transport you large distances from the ship can be risky. You can always do it cheaper but unless you are an experienced traveller or it is a well developed destination like Singapore or Hong Kong where everyone speaks English and it is easy to get to and from the docks (by the way the Diamond will be in one of the commercial dock areas)then it is probably best to do the tours.
IF going on your own, do some research on what you would like to see, get a group together and hire a cab, negotiate price first. Ask on cruise boards about specific tours and you will probably get some good advice from people who have done that specific tour or know that region.
Princess is a great cruise line, no matter what you will have a simply great time, write us a review when you come back...dont forget to take up the offer of a future cruise credit while on board. I guarantee you will be wanting to cruise again in the near future.