The only horcrux in this story is the diary, just to alleviate confusion. Pat yourself on the back if you catch the Pratchett reference. :)
Harry snuck into the garden and opened the letter. So, a wizard school. Awesome. He just had to get some time alone in the school library where they kept the maps. How nice of them to print the return address on the letter.
He packed his set of spare clothes and took fifty pounds from his aunt's purse as wages for the chores he had done over the years. One bus map of the UK and a sad story about being newly orphaned and going to his relatives in Scotland later, and he was on his way to Hogwarts.
True, he was a month early for school, but the arrival of a thin and horribly dressed boy made the few staff who were there take notice. Professor McGonagall dragged him in for a checkup as soon as she met him and what they found there had her up in the Headmaster's office, yelling at him in Scottish, for nearly five hours. At the end, a defeated-looking Albus Dumbledore slunk away, the castle having rejected him and his phoenix clearly abandoned him (Fawkes could make an impressive variety of rude noises and gestures for a bird) and she was wading into the paperwork to make the school year more safe and less complicated than he had planned it to be, eagerly assisted by Amelia Bones and Augusta Longbottom. They had been called in somewhere around Hour Two.
Harry met Professor Snape in the hospital wing and, when insulted in every possibly way, calmly asked for details of his parents' deaths, since he now understood that most magicals did not even own cars and so his surviving a crash caused by his parents didn't seem to be anything to be that terribly famous for, though it was surely rather unique. The professor stared at him blankly for a few seconds, used legilimency to see that he was completely sincere, and then told him an extremely short version of the true events and stomped off to complain to McGonagall.
The Philosopher's Stone was quickly returned with thanks to Flamel and the incumbent Defense Professor was asked to remove his turban, stunned and examined when he refused, and then arrested, de-possessed, and interrogated. Unfortunately, the thing that had been possessing him escaped. Hagrid had Fluffy returned to him. The castle staff learned some impressive new swearwords from the Headmistress.
At the Sorting when the school year finally began, the Hat argued with itself for half an hour between Hufflepuff and Slytherin for Harry, who sat there patiently. He finally told it point-blank that while he would like to make friends, he would like his friends not to be the kind who were friendly just because they were friends with everyone, and the Hat put him in Slytherin, but not before treating Harry to a rant about how irritating the old Headmaster was with his confunding spells to put his personal favorites all in Gryffindor. Harry listened patiently, sympathized, and tucked the mental note 'Albus Dumbledore, do not trust' away for later.
He was a little amazed by the spectacular tantrum a blond kid threw after he was sorted into Gryffindor when the Hat had barely landed on his head. He decided to keep his distance from the skinny pale version of Dudley. Also from the freckled redhead version of Dudley. Luckily they were both in Gryffindor, so it was easier to watch for them.
Harry's first year was quiet and full of learning. He made friends with a shy Hufflepuff and a competitive Ravenclaw as well as with several students in his own House.
He expanded his circle of friends in his second year, and nothing much happened until one day he stumbled up to the Headmistress' office clutching a ripped diary in one hand and a basilisk fang bigger than his arm in the other and apologized for temporarily borrowing Gryffindor's Sword. But he had gotten rid of the vandal that had been writing on the walls and making his friends miss weeks of classes, so he thought it was for a good cause.
His third year was even less exciting. His Ravenclaw friend's cat brought him the redhead Dudley's rat, and Fawkes, Harry's near-constant companion, told him that the rat was a man. He brought all three animals to his Head of House during the appointed office hours and quietly asked for a test to be run on the rat. When Professor Snape saw that the rat was a man, he nearly had a heart attack, scooped Harry up and bundled him into his private quarters which were the most strongly-warded area he could think of at short notice, and told him not to leave until he was fetched by Fawkes or the Headmistress. Harry spent his time playing Exploding Snap against both Crookshanks and Fawkes. Crookshanks was quite good.
He was pleasantly surprised to find out that he had a godfather, but since the man had been in prison for half his life and (Harry suspected) had had some developmental issues before that, he politely refused to live with him until the man had had some intensive counseling. For some reason, that made Professor Snape snicker.
He was annoyed when he was entered into some strange competition in fourth year, but calmly complied with the rules. For each task, he politely stood by when it was his turn until the timer expired and then walked away. The crowd was confused. Slytherin and Hufflepuff gave him a standing ovation each time. They liked his way of thinking.
No one that had spent time with him at Hogwarts was too surprised when he appeared out of nowhere at the end of the third task holding a dead baby-like creature. The other Hogwarts champion, Cedric, was holding the rat-man from second year. They both looked rather irritated. It had been terribly unsporting of Voldemort to interrupt their competition like that.
Far away in a hotel, Albus Dumbledore was horribly disappointed to read in The Prophet that when interviewed, Harry Potter had stated that ''the power the Dark Lord knows not was apparently a combination of critical thinking skills and civil disobedience, both of which I see are skills almost no wizard knows''. He then realized that someone had told Harry the prophecy and had a fatal heart attack.
Harry later married Luna Lovegood, who was merely a logical thinker and not crazy. He became the Head of Slytherin after Professor Snape became Headmaster, and Luna was the youngest Chief Warlock of the Wizengamot ever elected. Together with Minister Susan Bones the four of them managed to drag the magical world into, if not the modern era, at least the era where people could use gas-lamps and didn't have to sit on sheep.
Harry, in his memoirs, recalled that the claiming of that fifty pounds from his aunt's purse for his unpaid wages was the defining moment of his life, the moment he realized that he could indeed have a voice, and what ultimately led him to defeat a Dark Lord.