  We describe a framework to construct tropical moduli spaces of rational
stable maps to a smooth tropical hypersurface or curve. These moduli spaces
will be tropical cycles of the expected dimension, corresponding to virtual
fundamental classes in algebraic geometry. As we focus on the combinatorial
aspect, we take the weights on certain basic 0-dimensional local combinatorial
curve types as input data, and give a compatibility condition in dimension 1 to
ensure that this input data glues to a global well-defined tropical cycle. As
an application, we construct such moduli spaces for the case of lines in
surfaces, and in a subsequent paper for stable maps to a curve.
