This is the first paper in a two-part series containing some results on dimension estimates for $C^1$ iterated function systems and repellers. In this part, we prove that the upper box-counting dimension of the attractor of any $C^1$ iterated function system (IFS) on ${\Bbb R}^d$ is bounded above by its singularity dimension, and the upper packing dimension of any ergodic invariant measure associated with this IFS is bounded above by its Lyapunov dimension. Similar results are obtained for the repellers for $C^1$ expanding maps on Riemannian manifolds.