If the Universe (space) is expanding on a cosmic scale, is it also expanding on a local one? In other words, is the distance between planets or even electrons also increasing?
No. Locally, the distances between atoms, planets, stars, or even galaxies within bound clusters of galaxies is not increasing on average. The distance between us and the Andromeda galaxy is actually decreasing.
It is only on the largest scales that the distances between clusters of galaxies are increasing. As you correctly state, by the "expansion of the Universe" we simply mean the fact that such distances are increasing on the largest scales. We ourselves live in a small bound group of galaxies, the Local Group. Relative to us cosmic expansion begins on scales about 7 million light years away. For observers living in rich clusters of galaxies the equivalent scale would be 30 or 40 million light years.
Most of the present volume of the Universe is found in almost empty voids where gravitational collapse to form dense structures never happened. In such void regions there are still electrons, protons and helium nuclei between which the distances are growing, and have been growing, for the entire history of the Universe.
So an equally correct answer to your question is that in most of the volume of the Universe the distances between elementary particles is increasing. However, in practice we only observe matter which is so dense that its gravitational attraction was strong enough to overcome the initial expansion of the Universe, reversing that expansion to collapse and form clusters of galaxies, and all the structures within them.
Actual clusters of galaxies form thin filaments, like spiders' webs which thread voids, and sheets which surround the voids. Knots of rich clusters of galaxies occur where dense sheets intersect. Our own Local Group of galaxies lies in a filament which threads the Local Void. Knots, sheets and filaments of galaxy clusters all expand at different rates; in the denser regions gravity has slowed down the expansion between galaxy clusters more. Understanding the properties of this cosmic web is one of the most important questions in cosmology. The geometry generated by such structures according to Einstein's equations is extremely complicated.
My own research is based on the hypothesis that the mystery of "dark energy" can be explained by reexamining fundamental questions in Einstein's theory relating to the variation of average expansion in the "cosmic web".
Professor David Wiltshire
Department of Physics and Astronomy
University of Canterbury