Jeff,

First we need to know, do you wish to fly for a company or airline, or do are you looking to purchase your own aircraft to make these transcontinental/international flights? 

I will give you some insight into your other questions.  There are only a few things to international flying that differ from flying locally.  It is just simply, more planning.  You just need to find airports of entry (those with Customs) that can review (if necessary) the plane and passenger documentation (Passports, possibly luggage, etc.).

Otherwise, it is pretty much the same (you will have to deal with different accents) and the charts/IAP's used may look slightly different, but no big deal.

The typical transition is piston-to-turboprop-to-jet.  The only difference is the complexity of the systems.  When you break it all down, they are all just airplanes and the 4 fundamentals of flight apply to all of them the same.

If you want to be a good pilot, learn the right way.  Once you have learned the right way, always do it that way.  Do not take short cuts.  Short cuts are bad habits that can be picked up along the way (not using a checklist, not doing a thorough pre-flight, etc) that will one day catch up to you.

If my experience has taught me anything, it is that I am always learning new things.  Maybe not "learning" things, but seeing and hearing things that I did not think were possible; 9/11 being the most notable.

Brian