[Grigory Mikhalkin][1] has a few papers which you might find motivational. 
To start, a survey on tropical curves from the AMS Notices "What is ..." column from April 2007, available here http://www.ams.org/notices/200704/index.html

Next, a longer survery article from the 2006 ICM, aptly titled 
[Tropical Geometry and its Applications][2], where he covers your three mentioned applications, extends the idea of curves in the previous article to general varieties, and then gives nice ways to compute Gromov-Witten invariants of $\mathbb{CP}^2$. If you like real algebraic geometry, he also computes Welshinger numbers, and methods of gluing hypersurfaces of toric varieties in higher dimensions.

Finally, the paper "Enumerative Tropical Algebraic Geometry," available on his website, gives a method for computing multicomponent Gromov-Witten invariants, as well as some other counting applications.

You might also like this question: https://mathoverflow.net/questions/53306/what-can-we-learn-from-the-tropicalization-of-an-algebraic-variety


  [1]: http://www.math.utah.edu/~gmikhalk/
  [2]: http://arxiv.org/abs/math.AG/0601041