Technically, my first was Gatchaman in the form of Battle of the Planets. I just had no idea it WAS anime because I was very little. But I just knew *something* was different about it from other cartoons.
Later on like many I also watched Voltron, but again I didn't really know about anime. Later on I found out about anime after I had gotten into Area 88 and Heavy Metal Warrior Xenon mangas (some of the earliest to be brought over to the U.S.) but still had not really seen any except what was dubbed or hacked during my childhood. Around that same time a buddy of mine in school found out about a anime club at Virginia Tech and we went to it one night.
They were running Record of Lodoss War, BubbleGum Crisis, Dangaio, and Ranma 1/2 long before any of those had been licensed over here. I don't remember which ran first but it was definately a awesome experience.