First things first – to say, „I have a meeting with my best friend“ sounds strange. It sounds far too official, like you have very important business matters to discuss. And that normally isn’t true.
So, how should we say it? A date? No, that is romantic. An appointment? No, that is formal and what you have with a doctor or optician.
Far more natural is to either use the verb ‘meet’ or ‘see’. So, you might say, „I am seeing/meeting my best friend“, or in the case of the above regular example, „I see/meet my best friend every Thursday.“