Unsure of what to think about negation of Continuum Hypothesis, a need of clarification It is written on Wikipedia:

In mathematics, the continuum hypothesis (abbreviated CH) is a hypothesis about the possible sizes of infinite sets. It states:
There is no set whose cardinality is strictly between that of the integers and the real numbers.

Suppose that CH is not true, that is that there is such a set, call it $C_1$, so, $|\mathbb Z|<|C_1|<|\mathbb R|$
But, does some principle "forbids" us to suppose that there are also sets $C_2$ and $C_3$ so that we have $|\mathbb Z|<|C_2|<|C_1|<|C_3|<|\mathbb R|$, and $4$ more sets with cardinalities between those $5$, and so on and so on...

That is, if CH is not true, can there be an infinitely countable number of sets $S_i$ between $\mathbb Z$ and $\mathbb R$ so that we have $|\mathbb Z|<| S_1|<...<|S_{\infty}|<|\mathbb R|$. And, if there cannot be an infinitely countable number of sets between $\mathbb Z$ and $\mathbb R$, all with different cardinalities, can there be a finite number of them (a finite number greater than $1$)?

 A: Easton's theorem in particular implies (although the result itself is probably easier to prove than the full theorem) that if you fix a model $V$ of ZFC+CH then for all cardinal $\kappa$ of cofinality $>\omega$, there is a forcing extension of $V$ that preserves cardinals and cofinalities where $2^{\aleph_0} = \kappa$ ( note that $|\mathbb{R}|=2^{\aleph_0}$). 
Now if $\kappa = \aleph_\alpha$, the "number of cardinalities between $\mathbb N$ and $\mathbb R$" is $\alpha$ (or "$\alpha-1$" arguably, but for limit $\alpha$ one can safely say $\alpha$). In particular, for any ordinal of cofinality $>\omega$, this number can be $\alpha$. 
Your last question in the comments is "can it be $|\mathbb R|$ ?", i.e. can we have $\alpha = \kappa$, that is, $\aleph_\kappa= \kappa$. This question is : is there a cardinal of cofinality $>\omega$ that is an $\aleph$-fixed point ?  The answer is yes, as for any regular cardinal $\kappa$ there are (many) $\aleph$-fixed points of cofinality $\kappa$, see for instance here for a proof. So in conclusion : there can be $|\mathbb R|$-many cardinals between $\mathbb N$ and $\mathbb R$ (of course not more)
