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.
As you state, "intrados" is the architectural term for the interior of a dome or of a flat arch. In casual conversation, I might refer to the "interior of the dome", or call it a "bowl".
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"