Leonardo and Michelangelo worked on commissions that were nearly side-by-side in the Palazzo della Signoria in Florence. At least a couple sources (including Leo himself, as well as the 16th-century artist biographer Giorgio Vasari) mention their interactions or attitudes towards each other. Vasari claims that Leo and Michelangelo had a "great disdain" for each other (http://members.efn.org/~acd/vite/VasariLeo3.html). Interestingly, Leonardo was also present at the meeting, in 1504, to determine the placement of Michelangelo's David upon that statue's completion (Leonardo favored placement in the Loggia dei Lanzi, against the wall).

I'm less aware of interactions between Raphael and the other two, although Raphael certainly was aware of the senior artists (and responded to their work in his own work). There are direct quotations of Leonardo and Michelangelo by Raphael (e.g., the seated female figure at the right of Raphael's Deposition is a mirror quote of the Virgin in Michelangelo's Doni Tondo; see here and here to compare).

The most interesting ways these artists "interacted" was via their work, which was often very public. The best source for this—and one that is highly readable, even for someone who isn't an art historian—is, in my view, Rona Goffen's book Renaissance Rivals: Michelangelo, Leonardo, Raphael, Titian (2002).

Another source probably worth looking at is William Wallace, “Michelangelo in and out of Florence Between 1500 and 1508,” in Leonardo, Michelangelo, and Raphael in Renaissance Florence from 1500 to 1508, Washington, D.C. 1992