Is $\mathbb{R}$ a subset of $\mathbb{R}^2$? Is it correct to say that $\mathbb{R}$ is a subset of $\mathbb{R}^2$? Or, put more generally, given $n,m\in\mathbb{N}$, $n<m$, is $\mathbb{R}^n$ a subset of $\mathbb{R}^m$?
Also, strictly related to that: what is then the "relationship" between the set $\{(x,0)\in\mathbb{R}^2,x\in\mathbb{R}\}\subset\mathbb{R}^2$ and $\mathbb{R}$? Do they coincide (I would say no)? As vector spaces, do they have the same dimension (I would say yes)?
If you could give me a reference book for this kind of stuff, I would really appreciate it.
Thank you very much in advance.
(Please correct the tags if they are not appropriate)

Strictly speaking, $\mathbb R$ is not a subset of $\mathbb R^2$.
However, depending on how you construct your number systems, the natural number $1$ is different from the rational number $1$, and both are distinct from the real number $1$.  And don't get me started on complex numbers or quaternions!
This type of imprecision allows us to actually talk about the objects in question without worrying too much about auxiliary details.  $\mathbb{R}$ can clearly be embedding into $\mathbb{R}^2$ preserving virtually any property of $\mathbb{R}$ in the process.  Once this is realised linguistic license is not too detrimental to rigour.
