Ok here I go...
1. Web Swinging! Spider-Man 1 & 2 had some great swinging scenes that were used to show Spider-Man's entrance/exit into/from a scene. We had very little swinging in Spider-Man 3. That always urked me, so use plenty of web swinging in the reboot.
2. Balance of Spider-Man/Peter Parker: I thought the first and second movies nailed that balance. Spider-Man 1's mid-late parts had scene after scene of Spider-Man either saving Mary Jane (three times

) and fighting Green Goblin. And there was breaks between those stints of Spider-Mania with some great parts with Peter talking to MJ/Harry/JJJ/Aunt May.
3. Humor: I don't want a lot of humor, like in Spider-Man 3, but I want Spider-Man dishing out insults too the villain(s). Not too much, obviously. A tad more than what Spider-Man 1 had. When the final battle comes, have no quips, but when it's just Spider-Man fighting the bad guy for the first and second time, have him do a lot of yapping. But if there's lives on the line, then I Spider-Man should shut up for a bit. But I want too see Spider-Man's humor come out more.
4. Character Bonding: Obviously, Peter can't fall in love right away. I think they should have Aunt May have a heart attack, and Peter and Gwen are taking care of her, at that time they start to bond and maybe start like each other more.
5. KILl GWEN STACY! My god, we have the perfect opportunity to kill her in the first/second movie. Please don't ***** out. I want to see GG kill her. We don't have to have a whole Goblin saga, but I do want to see GG in better suit this time
6. Don't kill the villain(s) unless necessary. This was one of my annoyances with Raimi's Spider-Man. At least save the villain so he can make cameos in future films. Maybe in later films they can have Spider-Man go to the jail to test the systems. **** goes down, and Spider-Man has to defeat the super villains that are stuck on the island prison with him. Sinister 6 maybe?
7. Final Swing. I always liked the Final Swings. If you buy Spider-Man 1 on Blu-Ray and put the Final Swing on a loop, you've got your money's worth. It's beautiful scene and I would love to see it again in the reboot.
8. Keep the damn mask on! I don't care if the suit gets torn up, but the masks stays on. And if the mask gets ripped. Make sure Peter's face is bleeding/bruised. In Spider-Man 3, Spider-Man's mask gets ripped and his face is perfectly fine. Even after multiple poundings from Venom and Sandman. It doesn't make sense!
9. At least one movie without a damn damsel in distress! Hell, for the first movie, I want to see Spider-Man stop the villain from doing something that could kill civillians or something like that. Let Gwen not be kidnapped for one movie!
10. Don't link the Villains to Peter! Now, guys like Normon and Conners I'll let slide, but don't link Electro or Shocker to Peter in anyway.