Though Gilbert Blythe's a major (and majorly evil) character in Anne's mind, we don't get to know the actual Gilbert that well. We know he's fair. Maybe he's even noble, considering he tried to take the blame after Anne literally smashed a slate over his head, and later, gave up his teaching position for Anne even though they weren't speaking.
But do those actions show his general nobility, or are they just clues that he like-likes Anne? Don't forget that Marilla used to be into Gilbert's father—another clue? But we're getting ahead of ourselves. Save that for the later Anne books.
Back to Gilbert's personality. We know he's smart. He teases the girls when he's younger, but befriends a bunch of them later—pretty typical. But we don't get to know any specific about him, because after the "carrots" incident, the only times he and Anne speak are when they're forced to.
If you go on to read the rest of the series, you'll get to know Gilbert better as a person, but in this book, it's all about what Gilbert stands for in Anne's mind: he's a worthy rival. Her desire to beat him pushes her to study more and to do better in school.
When Anne finds out Gilbert won't be going to college with her, she's sad, even though they haven't made up yet:
Anne felt a queer little sensation of dismayed surprise… What would she do without their inspiring rivalry? Would not work, even at a coeducational college with a real degree in prospect, be rather flat without her friend the enemy? (36.26)
Gilbert spices up Anne's academic life, and her work would be "flat" without it. With descriptions like "inspiring" and "her friend," the narrator shows us just how important Gilbert has become to Anne.
And they haven't even smooched yet.