Prove that for every planar graph, there is a partition $V = V_1 \cup V_2 \cup V_3$ such that the graphs with those are acyclic 
Prove that for every planar graph $G = (V,E)$ with $|V| \geq 3$ there is a partition of V to $V = V_1 \cup V_2 \cup V_3$ such that $V_1 \cap V_2, V_1 \cap V_3, V_2 \cap V_3 = \emptyset$, where for all $1 \leq i \leq 3$ the graph that is formed by $V_i$ is acyclic.

I am still not able to comprehend the "tricks" that are used to solve Graph-Theory questions, and I am having a tough time with them.
When I read the question, I said this:

If $G$ is planar, then all subgraphs of it are planar. That means that $V_i$ is planar for all $1 \leq i \leq 3$.

But I still can not get a direction on how to prove that it is acyclic. Any direction would be appreciated.

Wayne Goddard, Acyclic Colorings of Planar Graphs, gives a proof of a stronger result, and a reference to a proof of the result you want. I think the paper is freely available on the web. The earlier paper is G. Chartrand & H.V. Kronk, The point-arboricity of planar graphs, J. London Math. Soc. 44 (1969) 612–616.
