Sets and Notation. There is a set like:
$V = \{f : R \to R \mid f(x) = ax + c \text{ with } a, c \in \mathbb{R} \}$
I do not know what "$:$" means.
I do not know what "$|$" means.
I think the meaning is something like "is defined as" but i can not find more documentation.
What is the difference between "$:$" and "$|$"?
When do i use the one and when the other?
How would i use them in a spoken sentence?

Edit: changed x to V.
Edit: changed example to make more senese in general
Edit: changed example to improve even further.
 A: It depends on the context.  Here is an example:
Consider both sets $\{ x \in \mathbb{R} : x < 2 \}$ and $\{ x \in \mathbb{R} | x < 2 \}$.
These sets both represent the same set.  The : and | should both be read as "such that".  So you read both sets as "the set of $x$ in $\mathbb{R}$ such that $x < 2$".

Now, when we define a function, say, $f: X \to Y$, the : is included after the name of the function, in this case $f$, and before the listing of the domain and codomain of the function.  So $f: X \to Y$ is how we start to define a function $f$ with domain (or input) from $X$ and codomain (or output) in $Y$.
The problem is in your example, $\{x : R \to R | x < 4\}$, we are using both : and |.  But the : looks like it is used as defining a function $x$ with domain $R$ and codomain $R$.  Then the | looks like it is a such that.  So, I would read this set as "the set of functions $x$ from $R$ to $R$ such that $x < 4$".
Now, does it really make sense to say functions are less than $4$?  No.  We usually say a function's output is less than $4$, so that makes me believe there is something wrong with the set you wrote down.  Is it possible you copied it down wrong?  But from a math language point of view, the set you wrote down can be read in plain English, as above.  It just doesn't appear to be a meaningful description.  
(Although, there is a debate as many mathematicians use $x < 4$ for a function $x$ as shorthand to say for all inputs, the output is less than $4$, but this notation is rarely used in undergraduate courses, from my experience.)

EDIT
Ok, so you updated the set description.  Everything I said above is still true, only now your new set is also mathematically meaningful. $\{ f : R \to R \mid f(x) = ax + c \}$ is "the set of functions $f$ from $R$ to $R$ such that $f(x) = ax + c$".  
As long as you know what $a$ and $c$ are, this makes sense.  If they remain variables, their description should be included in the set -- e.g., are they real numbers, or complex, etc?  If real, the set should look like $\{f : R \to R \mid f(x) = ax + c \text{ with } a, c \in \mathbb{R} \}$.
