What are some millitary innovations of the ancient Greeks?Please include the Athenian and Spartan accomplishments.

That's an excellent answer.  Of course Greek Fire is probably the most famous of their innovations, although the exact formulation(s) has been lost through the years.  The development of the phalanx as a formation in combat maximized the advantages of their armor, helmets, and shields, and worked well in close combat with the spear (the primary offensive weapon) and the straight and curved short swords, the xiphos and makhaira.  The blades of their weapons, of course, made from bronze were prone to bend, and quite different from the iron weapons of later times and those iron weapons used earlier by such cultures as the Hittites, and others in the areas of Palestine and the Levant.

Interestingly, some of the most innovative of Greek developments were in the areas of tactics and strategy, especially in their uses in naval warfare.  The naval victory at Salamis was perhaps the best example.  I suggest study of The Peloponessian War, an eyewitness account of the first "great war" of history, the 23-year struggle between the Athenian and Spartan alliances, written by Thucydides, a general during the war.