If I were going to a new confessor (and thank God I’m not in that situation now), I would indicate my approximate age and state in life, tell how long it has been since the last confession, tell my mortal sins by kind and number and any venial sins that I’m really sorry for and struggling to overcome. I might include a sentence about what I see as the cause of the sins, if I understand it myself, in order to give the priest an idea how to help me.
Like Joannm, I now have a regular confessor who knows me pretty well (it hasn’t been that long with him), so I just say, “Hi, Father, it’s Betsy.”
The most important thing for a good confession, however, is none of this stuff, but sincere contrition and purpose of amendment. Those aren’t part of the “script,” but in the heart and absolutely essential.