First, your question.
Yes, the increase of mass occurs only when a particle approaches $c$ (speed of light in vacuum).
$c$ is fundamental in Special Relativity, not because it is the speed of photons, but because it is the constant speed in the universe (the only speed invariant to boosts). Just because macroscopic light is transmitted at a lower speed inside a particular medium, that doesn't mean that the fundamental speed of Special Relativity is any different. Even inside mediums where light travels more slowly, all relativistic effects happen when a particle approaches $c$.
Since Cherenkov radiation (CR) is just an effect related to the speed of light in a medium (and not to $c$), it doesn't have anything to do with mass increase either. Though CR and mass increase can happen simultaneously to a particle, they are independent (the first does not imply the second, and vice-versa).
Second, about the increase of mass.
It has been a historical habit to say that a particle's mass increases as $m=\gamma m_0$ when its velocity approaches $c$. That is not very appropriate. While it may seem convenient to define this relativistic mass, it's not a good habit.
First, because it's confusing to some people. There are physically intuitive ways to explain to a student why time intervals must stretch and why space intervals have to contract, but there's no way at all to explain why a particle's mass should increase.
Second, it's also not accurate. The defined relativistic mass parameter does not sustain the properties you would expect from a mass under close analysis. (I have a reference for this, but the pdf file somehow got corrupted in the last 8 years. I'm looking for a copy.)
I've heard that a spacecraft could never exceed the speed of light
because it's (relativistic) mass quickly approaches infinity and
therefore there could never create a big enough rocket to propel it
faster and faster.
In fact, the spacecraft could never even reach, much less exceed the speed of light.
I think that you'll agree that the spacecraft, no matter what speed it may have relative to some other object, is at rest with respect to itself.
Think carefully about that! The spacecraft (or any material object) is not moving with respect to itself.
This seems so intuitive, so unquestionably true that you might think that there is no reason to even mention it.
But, according to Special Relativity, something that moves with speed c (the speed of light in vacuum) relative to some other object, moves with speed c relative to any object (c is an invariant speed).
In other words, if it were the case that the spacecraft could obtain the speed of light, it would be moving at the speed of light with respect to itself.
This is so plainly incomprehensible that it is, in fact, a relief to know that the spacecraft can never attain the speed of c.
More precisely, according to the Lorentz transformations, there is no frame of reference that moves with speed c relative to another frame of reference.
More generally, this fact "shows up" as nonsense statements like "the mass is infinite at the speed of light" or "infinite force is required to get to the speed of light".
There's no such thing as infinite mass or infinite force which is to say, you can't get to c from here.
Best Answer
The mass (the true mass which physicists actually deal with when they calculate something concerning relativistic particles) does not change with velocity. The mass (the true mass!) is an intrinsic property of a body, and it does not depends on the observer's frame of reference. I strongly suggest to read this popular article by Lev Okun, where he calls the concept of relativistic mass a "pedagogical virus".
What actually changes at relativistic speeds is the dynamical law that relates momentum and energy depend with the velocity (which was already written). Let me put it this way: trying to ascribe the modification of the dynamical law to a changing mass is the same as trying to explain non-Euclidean geometry by redefining $\pi$!
Why this law changes is the correct question, and it is discussed in the answers here.