I think your instinct is that 'appointment' is for something more formal, right? Well, your instinct is correct. 'Appointment' is for something more official. You have to make an appointment to see a doctor or the dentist because there is a limited amount of time available for them to see patients. You wouldn't use it to describe meeting up with your friends, it implies a professional distance between the people. I'm going to meet my friend. I'm going to see my friend. I'm going to meet up with my friend.

"Appointment" does work, yes, but you're right that it's farther along the formality spectrum than alternatives such as "I'm meeting my friend", "I'm going to get together with my friend". None of them sounds wrong. But it's possible that the words you use will send subtle subconscious signals to the listener about the true nature of your relationship with the friend. A wife would probably not want to overhear her husband saying, "I'll call you back. I just arrived at the bar and I have an appointment with my wife now". :) So it depends on the circumstance as to how important the word-choice is.

If it is with a wife or girlfriend some people will use date. Other than date though there really isn't a set noun for an informal meeting. Generally we just describe what we'll be doing such as "I'm going to go walking with a friend." If you want to be nonspecific something like "I'm going to meet someone"