I just started learning some basic group theory. The definition of cyclic groups I read is as follow:
A group $G$ is a cyclic group if there exists some element $g\in G$ so that $G=\{g^k\,|\,k\in \mathbb{Z}\}$.
The material I read gives an example of $G=\Bbb{Z}_6$, and then its cyclic subgroup is $\langle 2 \rangle=\{0,2,4\}$. But according to the definition, only 2 and 4 would be in $\langle 2 \rangle$ because $2=2^1$ and $4=2^2$. Why is 0 in there but not 1 (since $1=2^0$ ) ?
You're mixing additive and multiplicative notation. So $2^0=0\cdot 2=0$. $2^1=2$, and $2^2=2+2=4$.
It's an easy mistake to make. Especially considering that $\Bbb Z_6$ is also a ring, so has addition and multiplication.
The identity of the additive group $\Bbb Z_6$ is $0$. It has to be in your subgroup.