I want to make a new game center account for myself, because I share one with my mom and brother. Does it delete your progress in games if you do?
The Game Center progress stays with the account, not the device so you would need to check with each specific application to see if they also maintain the game status and would propagate it to a new account or you would be starting over when it detects a new account has been added to the same device.
There isn't a general "it always works this way" established in the games I have tried. Some certainly have their own data servers and data structures so you don't lose progress without a game center account. Others depend entirely on Game Center and only the account stores any sort of progress and achievements so that all status is 100% cloud based and not reflected on the device as a permanent store of progress.