Understanding random variables I am just getting into probability theory, and don't quite understand the concept of random variables. It is presented to me as a transformation, it takes input from a background space of "events" and gives output in a space of, say, real numbers. This makes intuitive sense, as we are simply "labeling" events in some way.
However, shortly after this introduction, the book ignores everything about the definition it just gave, and starts talking solely about "distributions" of random variables, and "distributions of transformations" of random variables. Almost every exercise now begins with "let $X$ be a blablably distributed random variable" as well ... 
What does it, intuitively, mean to have some distribution, and why is that more important than knowing the explicit definition of $X$ (say, $X(x) = |x|$, or something)? I understand that knowing the distribution of $X$ is useful, ... but don't we want to go in the other direction instead? Start with $X$ and figure out its distribution, instead of defining distributions and just using them?
 A: Let me trace this backwards, starting from your last questions. Essentially, what you are touching upon is the fundamental difference between statistics and probability theory. In probability theory, we work with variables from some distribution, and derive properties from them.
In statistics, however, the distribution is usually unknown. We only have the data, and from there we try to infer a distribution. In some sense, this is probability theory turned upside down!
A random variable having a distribution essentially is an assignment of probabilities to specific values of said variable. You need to understand that a random variable is random: its 'true' value is not known, but we do know how those values are distributed across some space (say, the real line) - conditional on the distribution. Based on that distribution, we can then infer how likely some events (data) are, as opposed to others - compare this to statistics, and you will see how elegantly the two relate to each other!
A: Ok, so you understand what a random variable is, but we want to deal with probabilities of events occurring.  This is where distributions come in handy.  The distribution of a random variable $X$ is a function $f_X(a) = P(X=a)$.  So, distributions record the probability that the random variable achieves a specified value.  Since we want to be able to answer questions about probabilities of random variable occurrences, distributions help us out.
For example, if we flip a fair coin 20 times, we could ask what the probability of achieving exactly 3 heads is.  To solve this, let $X$ be a random variable recording the number of heads in 20 flips of a fair coin.  Then the distribution of $X$ is Binomial with parameter $1/2$.  Using this, we can rephrase our question as "Compute $P(X=3)$."  Then knowing how Binomial random variables are distributed allows us to say $P(X=3) = f_X(3) = {20 \choose 3}(\frac{1}{2})^{3}(\frac{1}{2})^{17} \approx 0.12\%$.  Knowing common distributions allows us to solve these problems with relative ease.
A: You should firstly get familiar with the most common distributions - eg. binomial, Bernoulli, geometric etc. - in order to be able at a later stage to identify by yourself the distribution of a given $X$. 
Now, very roughly random variables are the analogous to functions when things are random. How difficult was it for you to grasp the concept of $f(x)$ at school? It will be a little more difficult to grasp the concept of random variable, trust me. 
Imagine we throw a dice and I tell that given the angle (a), the temperature (t), etc. etc. we can determine the result and say that it will be given by $f(a, t)$. Say $f(a,t)=at^2+3a+\sin t$ (purely hypothetical). This is a function - as you have learned it -and you know what to do with it. That is a deterministic experiment with input some variables and output the result of the dice. Up to here ok. 
But the result of the dice is random and in fact we cannot predict it given the above data and there no such (known) $f$. Instead we say that $X$ (instead of $f(a,t))$ is a random variable which takes values in $\{1,2,3,4,5,6\}$ (recall that we said about functions, that they had a domain and an image) with the "law" or "distribution" $1/6$ each (that is the analogous to the type of $f(a,t)$ say $f(a,t)=at^2+3a+\sin t$. Now there is no such formula but the "distribution"). 
Now, the common definition that $X:Ω\to \mathbb R$ is a little bit confusing at the beginning indeed, so be patient and you will come back to it later.
