The need to evaluate expressions of the form f(A)v or v'f(A)v, where A is a large sparse or structured matrix, v is a vector, f is a nonlinear function, and ' denotes transposition arises in many applications. Standard and rational Krylov methods can be attractive for computing approximations of such expressions. These methods project the approximation problem onto a standard or rational Krylov subspace of fairly small dimension, and then solve the small approximation problem so obtained. We pay particular attention to the case when A is symmetric and the rational functions have poles at the origin and at infinity. Then an orthogonal basis for the rational Krylov subspace can be generated using short recursion formulas. These formulas are derived using properties of Laurent polynomials. We also will discuss the approximation of expressions of the form v'f(A)v with block Gauss quadrature rules. Applications to network analysis will be presented.