Are these sets countable or uncountable? I get that to prove a set is countable you have to find a bijection from that set to a countable set, but i usually get stuck trying to find that function.
I'm trying to prove whether or not these sets: $\mathbb{Z}^{[0,1]}$ and $[0,1]^{\mathbb{Z}}$, are countable or uncountable. 
Thanks! 

I hope that by $[0,1]$ you mean the interval and that it is not a typo for $\{0,1\}$.
Finding an explicit bijection can be (very) hard. Instead, one can use the CSB theorem to show two sets have the same cardinality by showing that injections in both directions exist. Or, to show that a set is countable, one can express the set is a countable union of countable sets. 
To show that a set $S$ is not counatable it suffices to find an injection $B\to S$ from an uncountable set $B$. This is possible your second set. Can you think of an injection $B\to [0,1]^\mathbb Z$ for a suitabley chosen $B$? Can you then solve for the first set? 
