Does the expansion of the universe increase with distance? I read something in a book the other day which has gotten me really confused. They were talking about the expansion of the universe, and if I understood it all, they said that the expansion of the universe increases with distance. I’m wondering if I understood this correctly, because I was reading about Hubble’s law and it said that the redshift if galaxies far away (caused by the expansion of the universe) is proportional to the distance (up to a few hundred megaparsecs away). Doesn’t this mean that the expansion is the same but the redshift increases? (I don’t see why the redshift would be proportional and a linear graph if the universe expanded from us proportionally to the distance, or at least more further away). I hope you understand what I mean by all this.
I'm not sure why you say that "expansion is the same" when you also say that "expansion increases with distance", but I wouldn't say the "expansion increases", but rather that "the recession velocity increases".

The recession velocity $v$ of galaxies increases linearly with distance $d$. The constant of proportionality is called the Hubble constant $H_0$. Thus, Hubble's law states that
$$
v = H_0 d,
$$
and currently, $H_0 \simeq 70\,\mathrm{km}\,\mathrm{s}^{-1}\,\mathrm{Mpc}^{-1}$, but in the past it was higher. That means that, for every Mpc (mega-parsec, $\simeq 3.26\times10^6$ light-years) a galaxy is from us, its speed away from us is roughly $70\,\mathrm{km}\,\mathrm{s}^{-1}$. This is the speed it has right now, but not the speed we see it have, since we see it in the past.

In fact, we don't really see a speed at all. We measure a galaxy's redshift $z$, and $z$ also increases with distance, but not linearly. Thats's because it depends on how much the Universe has expanded since the light was emitted, and the expansion rate changes with time. But for small enough distances, i.e. for galaxies that are not seen that far into the past, the relation is close to linear, so that $v = cz$, where $c$ is the speed of light.

This is indeed approximately true out to a few hundred Mpc. For larger distances, the linear relation becomes inaccurate, and one must instead apply a cosmological model of the expansion history of the Universe, which depends non-trivially on the densities of its various constituents.

Acceleration of the Universe

The term "accelerated expansion rate" refers not to the value of $H$, but to the rate at which the scale factor $a$ — the "size" of the Universe — increases (the Universe may or may not be infinitely large; $a$ is normalized such that, today, $a=1$). 

For a given value of $H$, we know the recession velocity of any two points separated by some distance at a given time. At a later time, these two points will have increased their distance, and hence — by Hubble's law — their recession velocity will have increased.

What was higher in the past is the Hubble parameter $H(t)$. That number will, in the future, asymptotically decrease toward a slightly smaller value of
$$
H(t\!\!\rightarrow\!\!\infty) = H_0\sqrt{\Omega_\Lambda} \simeq 56\,\mathrm{km}\,\mathrm{s}^{-1}\,\mathrm{Mpc}^{-1}.
$$
The Hubble parameter is defined as $H\equiv\dot{a}/a$, where $\dot{a}\equiv da/dt$ is the Universe's change in size per unit time. If $H$ is constant, and $a$ increases, then $\dot{a}$ also increases. That is, the increase in velocity of any two points accelerates. In fact, we can solve the differential equation and get that, in the future,
$$
a(t\!\!\rightarrow\!\!\infty) \propto e^{Ht},
$$
i.e. the scale factor increases exponentially.

The figure below shows the evolution of $a(t)$ until $t\sim130\,\mathrm{Gyr}$. On the left is a zoom-in of the first $20\,\mathrm{Gyr}$ where you can see how the Universe decelerated the first $\sim10\,\mathrm{Gyr}$, until dark energy started taking over the dynamics.



At the moment the acceleration is actually quite modest, but it will increase without bounds in the future; for instance, in 50 Gyr from now, $a\sim10$, i.e. galaxies will be $10\times$ farther from each other, but in 100 Gyr we have $a\sim150$.