The expansion of the universe doesn't involve anything travelling - it happens via the addition of more space in between all the objects. Everything ends up further away from everything else, but nothing actually moved through space. The amount of new distance that gets added then depends on how much distance is already there, so if you consider something far enough away, the amount of new distance appearing between you and it is more than light can travel across in the same time period (so any light trying to make that trip will never actually arrive).
It also doesn't have any effect on objects that are tightly bound to each other by gravity (or other forces). In the commonly used analogy of a balloon being blown up beneath our feet, objects connected by gravity are like a sticker fixed to the balloon; the outward "push" of expansion is too slow/weak to overcome their hold on each other. That's true of our galaxy as a whole, and I think of the whole cluster of galaxies in which we reside, so the appearance of new space is only actually visible on the largest scales. At least for now; if the rate of expansion accelerates then nearer objects might start to be "carried away", until eventually they end up too far away to be part of the Observable Universe.
Also noteworthy, "The speed of expansion" is a misnomer, because it's not a speed. A speed would be some amount of distance per time, but the unit used to describe expansion is the "km per second per megaparsec" - a distance per time per distance (i.e. every second, where before there was 1 megaparsec of distance between two objects, there is now a megaparsec plus some number of additional kilometres). But that's kind of a weird unit because it's distance / time / distance... which cancels out to just 1 / time. Which brings us on to the fun fact; when you measure the actual value of the rate of expansion, dividing one by that value gives you an approximation of the age of the universe (it's not exact because the expansion hasn't been linear)