Besides the "Oh woe is me" and "Romantics screw up" things he's shows concstantly(to add to that no marriage aspects period). I for one would add a desire to improve his fighting style because Spidey could explore the "Way of the Spider" like he has and in a sense as a nerd he'd would want to stand up for himself more if Spider sense doesn't always save him in physical combat. I'd like to explore his insight of thinking how Spider-Man could really help the world instead of just the streets of new york and how to make it possible(you know give Peter actual dream to persue for not only his self but to everyone who like him went through the ropes of life and needed hope to live on).
-PS: Seriously please no marriage responses, because I know thats what some of you are gonna say.