I would guess either The Doomsday Machine or Balance of Terror. There are a few exciting battles in a few other episodes, like Errand of Mercy and Journey to Babel, but for full-on space action- the two I mentioned have the most starship battle scenes. Doomsday Machine being about a planet-killer (more or less a self-automated Death Star that's shaped like a giant cone) that threatens to destroy and consume the most densely populated area of our galaxy. Enterprise encounters the battle-torn U.S.S. Constellation and learns of the threat. Things get hairy when Constellation's unstable and revenge-driven commander comandeers the Enterprise and launches an attack. Lots of phaser fire. Balance of Terror is basically a submarine story with the Enterprise playing cat and mouse with a cloaked Romulan Bird of Prey. Very exciting drama as Kirk battles wits with a champion Romulan commander (played by Mark "Sarek" Lenard). We even get a rare glimpse at the inner workings of the phaser control room in this episode.
Both are great episodes and the outdated effects of the 60's in no way diminishes from the action. The remastered episodes with new effects shots enhance it for reality's sake, but the episodes really don't gain any more excitement than what was originally done- the story, charaters, and production quality made these battles seem epic even in the old days. These episodes introduced the space battle for that generation, certainly nothing like it on TV. Prior to this, space battles were Flash Gordon-like kiddie fare. Star Trek made the space battle concept thrilling and epic.