On what scale does the universe expand? According to the theory (or my understanding of it), the universe is not only expanding, but speeding up.

If the galaxies are moving apart, are the solar systems within them also moving apart from one another (&amp; the planets within them), if they are, won't the Sun be moving away from the Earth?

Does the expansion of the universe occur at all scales or just large (inter-galactic) scales?
The short answer is: gravitationally bound system will not be ripped by the accelerated expansion. 

A longer answer: The current standard model ($\Lambda$CDM model) says that everything started with the big bang. It released a lot of energy that pushed the Universe and is since then expanding. While it's expanding, gravitational attraction has been working to create the structures that we see today (planets, stars, galaxies, cluster of galaxies). 

As the Universe expands, the density of matter and radiation within the Universe decreases. At about 5 billions years after the big bang, these densities became smaller than the density of Dark Energy (Dark-energy-dominated era). The strange things about Dark Energy are that, 1) its  density remains a constant regardless of the expansion of the universe, and 2) it has anti-gravitational properties - it pushes rather than pulls. Once Dark Energy came to dominate the density of the universe it began to dominate its overall dynamics, and has caused the expansion of the universe to accelerate.

But in structures which have already formed, gravity is much stronger than dark energy and so they don't feel it.