Arguments
Legolas and Gimli's favorite friendship-building activity is having benign arguments about things that have no real right answer.
Take the arts, for example. Elvish art—in clothing, architecture, painting, etc.—is flowing and asymmetrical, echoing the naturally meandering shapes of trees and rivers and the lay of the land. Dwarvish art, meanwhile, is all geometric and mathematical, hard points and level edges and perfectly measured angles. Elvish poetry is musical and fanciful; Dwarvish poetry is prosaic and adheres to perfect rhythm and rhyme.
There's no right or wrong answer to any of this, of course, but Legolas and Gimli will spend hours debating the virtues of their individual sides. Neither will change the other's mind by the end of it, but they'll both learn a lot about each other and their cultures, and even though they'll each put on a show of being quite fed up with the stubbornness of the other, both of them will secretly walk away quite satisfied with the conversation.
(They've given up on asking Aragorn to settle the arguments for them. He will always refuse to comment. But he likes to listen in on the debates, sitting in the corner and smoking his pipe.)