Meaning of times in $\mathbb{R}^m\times\mathbb{R}^n\rightarrow \mathbb{R}^{m\times n}$? I use $\mathbf{a} \times\mathbf{b}$ for the cross product, $\mathbf{a}\cdot \mathbf{b}$ for the dot product and $ab$ for normal multiplication ($a,b$ are scalars).
However, what is the meaning of times in $\mathbb{R}\times\mathbb{R}\rightarrow \mathbb{R}^2$?
Or $\mathbb{R}^m\times\mathbb{R}^n\rightarrow \mathbb{R}^{m\times n}?$
Is it the cross product?
Is it the dot product?
Is it normal multiplication?
Update:
Does these have any meaning
$\mathbb{R}^m\cdot\mathbb{R}^n\rightarrow \mathbb{R}^{m\cdot n}$ (the dot product)?
$\mathbb{R}^m \mathbb{R}^n\rightarrow \mathbb{R}^{m n}$ (normal multiplication)?
 A: It is a cartesian product. If $A$ and $B$ are two sets, then $A\times B$ is by definition the set of couples $(a,b)$ with $a$ in $A$ and $b$ in $B$.
In your case:
$$\mathbb{R}^m\times\mathbb{R}^n=\{(x,y);x\in\mathbb{R}^m,y\in\mathbb{R}^n\}.$$
A: I think, more generally, you need guidance on the notation $f\colon A \times B \to C$, which is the notation for "a function called $f$, from the Cartesian product $A \times B$ of sets $A$ and $B$, to the set $C$" (the Cartesian product $A \times B = \{(a, b) : a \in A, b \in B\}$ is the set of all ordered pairs with things taken from $A$ and $B$). More generally, the format is
$$\text{function name} : \text{domain} \to \text{codomain}$$
But this notation $f \colon A \times B \to C$ often says nothing about what the function actually does to pairs $(a, b) \in A \times B$ to produce some $f(a, b) \in C$, unless $f$ happens to have a particularly descriptive name/symbol. In this case, you'll often see functions introduced in "two parts",
\begin{align*}
f \colon A \times B &\to C \\
(a, b) &\mapsto \text{however $a, b$ determine $f(a, b)$} 
\end{align*}
where the first line specifies the function name and all the sets we need, and the second line actually tells us what $f$ does to the pairs $(a, b)$ (and note the new symbol $\mapsto$, which is used like $\to$ above. But $\to$ is used with sets, the domain and codomain, while $\mapsto$ is used between the actual input and output, to explain what happens to elements in the sets).

So you'll never see notation like $\Bbb R^m \cdot \Bbb R^n \to \Bbb R^{m + n}$, with the function placed between sets. Instead, you'll see the function name/notation in the place of $f$, put before the domain. So things like
$$
\cdot \colon \Bbb R^n \times \Bbb R^n \to \Bbb R
$$
is a (mildly confusing) notation saying that there's a function called "$\cdot$" that takes two vectors in $\Bbb R^n$, and gives you back a real number (we can assume it's the standard dot product).
$$\times \colon \Bbb R^3 \times \Bbb R^3 \to \Bbb R^3$$
would be the (somehow more confusing) notation to say there's a function called "$\times$" that takes two vectors in $\Bbb R^3$ and returns another vector in $\Bbb R^3$; probably it's the standard cross product on $\Bbb R^3$.
For a slightly-less-weird-looking example, we might use
$$
+ \colon \Bbb R^n \times \Bbb R^n \to \Bbb R^n
$$
to say that we have an operation called "$+$" that takes pairs of vectors in $\Bbb R^n$, and returns a single vector in $\Bbb R^n$ (and unless it's stated otherwise, everyone would assume "$+$" means exactly what you think it means).
The domain and codomain can come in all sorts of varieties. For example, functions don't have to be defined on pairs of things, in which case our domain isn't going to be a Cartesian product. So to talk about the standard square root function, we might write
$$
\sqrt{\ }\; \colon \Bbb R_{\ge 0} \to \Bbb R_{\ge 0}. 
$$
Or maybe we're handed a function with a fairly cryptic name, 
\begin{align*}
\operatorname{ev} \colon M_{n \times n}(\Bbb R) \times \Bbb R^n &\to \Bbb R^n \\
(A, \vec{v}) &\mapsto A\vec{v}
\end{align*}
but with practice, we can see it's the evaluation map that takes an $n \times n$ matrix with real entries and a vector in $\Bbb R^n$, and applies $A$ to $\vec{v}$.
A: None of the above on the left. That $\times$   is what appears between two sets to denote their cartesian product - the set of all ordered pairs whose first (second) element is from the first (second) set. 
The $\times$ on the right is ordinary multiplication.
Edit: 
The vector space on the left has dimension $m+n$, the one on the right has dimension $mn$, so the arrow can't represent an isomorphism. It might an injection. One possibility is that you're thinking of $\mathbb{R}^{m\times n}$ as the space of $m \times n$ matrices. Then the arrow could mean the function $f$ given by
$$
f(v,w)_{ij} = v_iv_j .
$$
(ordinary multiplication of real numbers onthe right). This map isn't an injection since $f(0,w) = 0$ for every $w$.
The vector space on the left is naturally isomorphic to $\mathbb{R}^{m+n}$. The arrow in
$$
\mathbb{R}^m \times \mathbb{R}^n \rightarrow \mathbb{R}^{m+n} 
$$
might then stand for that natural isomorphism. Perhaps that's what you meant to ask about. That's what your example when $m=n=1$ suggests.
A: $\Bbb R \times \Bbb R$ denotes the Cartesian product.  An element of $\Bbb R \times \Bbb R$ has the form $(a,b)$, where $a$ and $b$ are both in $\Bbb R$.
Basically, $\Bbb R \times \Bbb R$ is just a longer way of saying $\Bbb R^2$.
A: The same symbol in different contexts gets different meanings. Math in this aspect is somehow like poetry; the same word gets different meanings in different contexts. 
The symbol "$\times$", applied to sets like $\mathbb{R}$, denotes the Cartesian product. If $X,Y \neq \varnothing$, then $X \times Y := \{ (x,y) \mid x \in X, y \in Y \}$. If $X, Y := \mathbb{R}$, then $X \times Y$ by definition is simply the usual Cartesian plane. It is defined that $X^{n} := \{ (x_{1},\dots,x_{n}) \mid x_{1},\dots,x_{n} \in X \}$. Now you know what the superscript of $\mathbb{R}$ means.
Note that $\mathbb{R}^{m} \times \mathbb{R}^{n} = \mathbb{R}^{m+n}$.
A: To answer your first question $X \times Y$ is the Cartesian product of sets $X$ and $Y$. This is all ordered pairs $(x,y)$ where $x$ is a member of the set $X$, and $y$ is a member of the set $Y$.
$$X \times Y = \{(x,y):x\in X,y \in Y\}$$
An example is $\mathbb R \times \mathbb R^2$ which is all ordered pairs $(x,(y,z))$ where $x \in \mathbb R$ and $(y,z) \in \mathbb R^2$. Of course this can be identified with $\mathbb R^3$ by the bijection $(x,(y,z)) \mapsto (x,y,z)$.
A more interesting example is $[0,1] \times S^1$ which is all ordered pairs $(t,\theta)$ there $t \in [0,1]$, i.e. $0 \le t \le 1$ and $\theta$ is a point in the circle. This product $[0,1] \times S^1$ gives a cylinder.
The word product is unfortunate, and has nothing to do with multiplication. The torus - which looks like a bicycle inner tube - is the Cartesian product of two circles: $T = S^1 \times S^1$.
Your idea of the dot product doesn't work for two reasons. Recall that the dot product takes two vectors (of the same dimension) and gives a number. Instead of it being a set, it is a mapping between sets ($\mathbb R^n \times \mathbb R^n \to \mathbb R$). It doesn't make sense to dot a vector from $\mathbb R^2$ with a vector from $\mathbb R^3$; the dimensions are wrong. Think about matrices: it only makes sense to multiply a $p$-by-$q$ matrix with a $q$-by-$r$ matrix to get a $p$-by-$r$ matrix.
Similar remarks can be made about your idea of normal multiplication $\mathbb R^m \mathbb R^n$. Take an example: Take $(1,2,3) \in \mathbb R^3$ and $(1,2)\in \mathbb R^2$. How do we get something in $\mathbb R^6$? (In a natural way that has meaning?)
