We discuss a method to calculate explicitly the Néron-Tate heights of certain tautological integral cycles on jacobians of curves defined over number fields. We obtain closed expressions for the Néron-Tate height of the difference surface, the Abel-Jacobi images of the curve itself and of its square, and of the symmetric theta divisors. As an application we obtain a proof, in the case of jacobians, of a formula proposed by P. Autissier relating the Faltings height of a principally polarized abelian variety with the Néron-Tate height of a symmetric theta divisor. We also obtain new results of effective Bogomolov-type (ie, effective positive lower bounds) for the essential minimum of many tautological cycles on jacobians.