Three notes:
(1) There was an earlier MO question on "Learning Tropical Geometry."
(2) Bernd Sturmfels is giving a series of three lectures on "Algebraic Geometry: Tropical, Convex, and Applied" at the MathFest conference next week.
(3) Most directly to your question, see the engaging and succinct 2006 2-page manifesto, "Ten Questions in Tropical Geometry," PDF download:
Question 1. Is it possibly true that every tropical variety is shellable?
Question 2. Give a good formulation for the moduli space of curves of degree $d$ and genus $g$ lying in $\mathbb{TP}^2$.
Question 3. Investigate matroid subdivisions.
Question 4. Classify all “root system polytopes”,...
Question 5. Compute the (positive) tropical flag variety $GL_4/B$ in its Plücker embedding.
Question 6. What do the face lattices of tropical polytopes look like?
Question 7. When does tropicalization commute with intersection?
Question 8. What is the best axiomatization of tropical oriented matroids?
Question 9. Is the tropical discriminant of a defective point configuration a subfan of the secondary fan?
Question 10. How can you tell if something is a tropical variety?
