The first thing I think we should decide on is what kind of school we want.

My personal preference would be to have characters sorted, or let them pick the house they will be sorted in.
Also, maybe allow students of any age to start in the first year. Like, instead of getting a letter at a certain age, they have to be "discovered" first. This would allow for characters to be normal until something triggers their powers to start developing. I think this would make things more interesting as well, because there could be some wise, older wizard in your class who has no idea how to do magic, but he gives the best dating advice, and he knows everything about growing herbs and vegetables and fruits.