Does the universe expand at the same rate everywhere in the universe? Specifically, I am wondering if some areas of the universe expand faster than other areas and whether the faster expanding areas diffuse the expansion through the slower expanding areas or does the expansion occur at a uniform rate throughout the entire universe.
What's outside the observable Universe, we can't say anything about, but averaged over large enough scales ($\gtrsim$ a billion lightyears), it does indeed seem to be expanding uniformly.

However, the presence of mass, or more generally energy, retards the expansion. This means that on the scale of clusters of galaxies, the Universe expands more slowly, and on the scale of galaxy groups, the galaxies' mutual gravitational attraction will prevent them from receding from each other. This is also why our galaxy, Solar system, planet, and bicycles will never get torn apart (unless the cosmological constant is not a constant).

Conversely, in mass underdensities, i.e. the huge voids between clusters and filaments of gas and galaxies, expansion is increased (relative to denser regions). In fact, it has been hypothesized that that the observed accelerated expansion of the Universe is not due to dark energy, but could be an "illusion" from accidentally living in the center of a huge underdensity (e.g. Zibin et al. 2008). More recent observations seem to rule out this possibility, though