When going on a plane for the first time, my heartbeat went really fast because I was scared, and I started taking deeper breaths going wide eyed. Is that an anxiety attack?

No, that is simple anxiety. You can be anxious without have an attack. An anxiety attack is so much worse. You can't take deep breaths. You feel like you can't breathe at all. I fly a couple of times a year and even I get a little anxious before take off. It is perfectly normal for many people.

So other than being afraid at first, how was the flight otherwise?