The first thing he ever gives is a Pumpkin Pasty.
It's not quite a present, but Harry shares his sweets with Ron when he meets him on the Hogwarts Express.
“Go on, have a pasty,’ said Harry, who had never had anything to share before or, indeed, anyone to share it with. It was a nice feeling, sitting there with Ron, eating their way through all Harry’s pasties and cakes (the sandwiches lay forgotten).”
- Harry Potter and the Philosopher's Stone, Chapter 6 (The Journey from Platform Nine and Three-Quarters
Also not quite a gift, but he gives Ginny Lockhart's books.
When Harry got Lockhart's books for free from him, he gives them to Ginny and decides he'll buy his own. I'm not sure whether this would count as a gift, since despite Harry needing them for school, Lockhart giving them to him made him feel awkward and he wasn't happy about the situation. Him giving them to Ginny was not just out of kindness, but also of a desire to be rid of Lockhart's unwanted gift himself.
“Staggering slightly under their weight, he managed to make his way out of the limelight to the edge of the room, where Ginny was standing next to her new cauldron.
‘You have these,’ Harry mumbled to her, tipping the books into the cauldron. ‘I’ll buy my own –”
- Harry Potter and the Chamber of Secrets, Chapter 4 (At Flourish and Blotts)
Harry buys Ron and Hermione Omnioculars at the World Cup.
Harry buys both of his friends Omnioculars at the Quidditch World Cup, when he sees Ron's disappointment at not being able to afford anything.
“Wish I hadn’t bought this now,’ said Ron, gesturing at his dancing shamrock hat and gazing longingly at the Omnioculars.
‘Three pairs,’ said Harry firmly to the wizard.
‘No – don’t bother,’ said Ron, going red. He was always touchy about the fact that Harry, who had inherited a small fortune from his parents, had much more money than he did.
‘You won’t be getting anything for Christmas,’ Harry told him, thrusting Omnioculars into his and Hermione’s hands. ‘For about ten years, mind.”
- Harry Potter and the Goblet of Fire, Chapter 7 (Bagman and Crouch)
Harry actually buys Ron a gift for their fourth-year Christmas.
Harry's first preplanned gift to someone is when he buys a gift specifically for Ron for Christmas during their fourth year.
“They’re only socks,’ said Ron, who had gone slightly pink around the ears, though looking rather pleased all the same. ‘Wow, Harry –’ he had just opened Harry’s present, a Chudley Cannon hat. ‘Cool!’ He jammed it onto his head, where it clashed horribly with his hair.”
- Harry Potter and the Goblet of Fire, Chapter 23 (The Yule Ball)