There are a number of reasons why the narrator takes his date out on a canoe. First and foremost, at fourteen years old, he's too young to drive. Even if he had access to a car he wouldn't be able to drive it. But the young boy is not deterred. He figures that there's another way he can impress Sheila on their forthcoming date: by taking her to the gig in Dixford by canoe.
The narrator's a natural on the water, and taking Shelia out on a date by canoe means that he will get to remain in his comfort zone for much of the night. Going on a first date can be a stressful experience at the best of times, but especially when the person you're dating happens to be as suave and sophisticated as Sheila Mant. Traveling up river in a canoe will make the narrator feel a lot more comfortable about the whole experience.
The young man probably thinks that Sheila's way out of his league already, so he knows that he'll need something special to make her take notice of him. Hence the canoe, which he proceeds to shine with loving care on the day of the big date. By the time that Sheila shows up, the canoe is looking nice and shiny, with cushions plumped up against the bow in case Sheila gets in a pensive mood and needs to have something to lean up against. The narrator's clearly pulled out all the stops to impress his date. All the more disappointing, then, that things don't quite go according to plan.