Question: What is the best way to check isomorphism between structures? For example, checking isomorphism between the next structures:
$$\langle \mathbb{N},+\rangle, \langle \mathbb{Z},+\rangle,\langle \mathbb{R},+\rangle,\langle \mathbb{R}^{+},+\rangle,\langle (0,1),\cdot \rangle$$
By intuition, I can see how $\langle \mathbb{N},+\rangle$,$\langle \mathbb{Z},+\rangle$ and $\langle \mathbb{R},+\rangle$ aren't isomorfic, but intuition isn't always the best way for checking isomorphism. Therefore, I be glad to see how you suggest doing such questions, since I am putting a lot of time on them. Thank you!

Answer: 
An isomorphism between two magmas (or semigroups, or monoids, …) $(A,\bullet\_A)$ and $(B,\bullet\_B)$ is a map $\phi\colon A\to B$ satisfying
$$
\phi(x\bullet\_A y) = \phi(x)\bullet\_B \phi(y)
$$
(so it is a *homomorphism*) as well as being bijective (making it an *iso*morphism).

In particular, when $(A,\bullet\_A)$ and $(B,\bullet\_B)$ are isomorphic (meaning there exists an isomorphism), then they have the same cardinality as sets. This means that $(\mathbb N,+)$ and $(\mathbb Z,+)$ can't be isomorphic to any of the three others, since these are countable while the others aren't.

Now $(\mathbb N,+)$ and $(\mathbb Z,+)$ have the same cardinality (there is a set bijection $\mathbb N\to\mathbb Z$), but they are not isomorphic: both have a unique identity element, which is $0$ in both cases (so they are monoids) but in $(\mathbb Z,+)$ every element has an inverse while in $(\mathbb N,+)$ only $0$ has an inverse. (I'm assuming $\mathbb N=\{0,1,2,\dots\}$, otherwise one doesn't even have an identity element.)

The same argument applies to $(\mathbb R,+)$ and $(\mathbb R^+, +)$, assuming that $\mathbb R^+$ denotes the *non-negative* real numbers.

Finally $((0,1),\,\cdot\,)$ can't be isomorphic to any of the others, since it is the only one having an element $a$ such that $a\cdot x=a$ for all $x$, namely $a=0$.

So, to answer your question: if you can find a property satisfied by one structure that isn't satisfied by the other, then they can't be isomorphic. If you can't find such a property, you might try to construct an isomorphism.
First, you need to find a bijection $\phi$ between the carrier sets.
In your example, $\Bbb N$ an $\Bbb Z$ are both denumerable and there is a bijection.
Likewise, there is a bijection between each to of the sets $\Bbb R$, $\Bbb R^+$, and $(0,1)$.

Second, you need to check if the equation $\phi(x+y)=\phi(x)\oplus\phi(y)$ for all $x,y$ holds, where $+$ is the operation in the pre-image structure and $\oplus$ is the operation in the image structure.