I think the inside surface is called "intrados" and the outer side is called the "extrados". However, these terms seems to be borrowed from the Italian language, so I was wondering if there are alternative words or phrases I can use. What are the best word or phrase to use in a conversation, because using "intrados" sounds a bit too weird for me, and it's probably not commonly use and would probably confuse people.
You are correct that "intrados" and "extrados" would not be recognised by 'Mr Average', though they are the standard term used by Architects and the like.
The inside of a dome would probably be referred to as a 'ceiling' by most people, or as I have already said: "the inside/outside of this/that dome"