If I were the Headmaster of Hogwarts
1. The Sorting Hat would have to go.
On the one hand, the houses have caused nothing but awfulness throughout the history of time. If Slytherin is supposed to be the nest of all evil, why continue to place only the shrewdest students in it? Still, I can understand that in a boarding school environment, a "house" can really be the equivalent of a family, and since the school is already built to accommodate four houses, they can stay. But I would ditch the Sorting Hat. Instead, I would allocate students into houses completely randomly - put them into alphabetical order and do a quick 1-2-3-4 count. This way we don't brand students "brave", "smart", "loyal" or "potentially evil" from the age of 11 - we just put them in a group and hopefully we don't have more dark wizards forming simply due to peer pressure.
2. Orientation Week!
Muggle-born witches and wizards (as well as those raised as Muggles) would start school a little bit early for an orientation into the magical world. On the final day of Orientation their parents would turn up as well to hear more about the school, and to see a brief performance rehearsed by the kiddies, where they show a few simple magic tricks they've already learnt from the staff or the prefects who help organise the event. The whole process would give the Muggle-borns a bit more confidence in their school year - they may be behind their peers in exposure to magic, but as a bit of a compensation they already know the teachers and are familiar with the grounds of the school and Hogsmead.
3. Parent Liaison Officer(s)
This connects to the Orientation idea, and specifically to the fact that it's a bit of a tall order to expect Muggle parents to just sign on the dotted line (do they even need to sign anything?) and let their kid spend the next seven years in a school they've never even heard of, entirely unable to apply for any further education whatsoever at the completion of said education. The PLO would schedule a meeting (at the parents' home) at the time their child is sent their letter. They explain what Hogwarts is, tell them a little bit about the magical world itself etc. The PLO would also keep in contact with the parents throughout the child's education, explaining what O.W.L.s were, what career opportunities their child had ahead of them and so on.
4. Non-magical subjects
It would be ludicrous to think that the home-schooled wizards or even the muggle-born normal-schooled ones would know enough about maths, history, French, home economics etc. on arrival to Hogwarts. It must really come in handy transfiguring matches into needles, I don't doubt that, but how about even some household spells? Or indeed, some down-to-earth reading, 'riting and 'rithmetic.
5. Parseltongue
Perhaps I haven't thought this through, but I would suspect having Parseltongue lessons as part of upper-level Defence Against the Dark Arts classes could come in handy. It's obviously easy to learn (Ron Weasley isn't exactly the sharpest pencil in the box), and if it really is the language of choice of all evil-doing, it would really be useful to have lots of non-evil people understanding it, too.
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Right, your turn. How would the school be different if you were Headmaster of Hogwarts?