While Lexington and Concord did happen first, they were only skirmishes. The battle of bunker hill was the first real "battle". Skirmish is defined by The Free Dictionary as "A minor battle in war, as one between small forces or between large forces avoiding direct conflict." Battle is defined as "An encounter between opposing forces." So, while Lexington and Concord fall under the category of both "battle" and "skirmish", skirmish would be the technically correct definition.
It is not the first battle of the revolutionary war. The first battle was the battle of Lexington Massachusetts. The battle of Bunker Hill was the first MAJOR battle in the war. However it was not the first. Trust me, I have a 97 in world history. I would know!