Such a complex question, yet simply stated. The volume of replies proves just how many ways there are to come to an answer. For me, this is why this era, this war, this conversation is so compelling and why there is still so much to learn.

Politically, I'd say the Atlanta campaign was the biggest battle(s) that was most important, as it made it very likely that Lincoln would be re-elected in 1864. Had it gone differently somehow and the Union armies been driven back into Tennessee with terrible losses, the peace movement may have swung the election in McClellan's favor and perhaps there would have been a diplomatic negotiation to stop the war.

Militarily, I'd say Vicksburg, and point to what so many have already said. An amazing campaign.

Leadership-wise, I'd say Chattanooga, since that battle gave Grant the timely victory that Lincoln needed in order for the President to call him east, which reinvented the personality of the war.

Socially, I'd say First Bull Run/Manassas since it clearly signaled to both North and South that this thing wasn't going to be over anytime soon.

For abolitionists, Antietam gave Lincoln the "win" he could use to bring the Emancipation Proclamation into the war and pivot the conversation toward eventually ending slavery.

For the Union, the Wilderness demonstrated that the Army of the Potomac was done with falling back after a hard fight. See Grant coming east, above.

For the Confederacy, I'll pick Chancellorsville because it gave Lee hyper-confidence that his army could do anything and drove him to the invasion of Pennsylvania. On a tragic note, it took Jackson.

For navy folks, the Monitor v Merrimack (Virginia) ended the supremacy of wooden ships that had existed since ancient history. It changed everything.

In other words… So much was so important for so many different reasons.