Use induction. Use the facts that if we remove a leaf from a tree it still remains a tree and a leaf is adjacent to only one vertex.
First, a graph is bipartite if it can be colored with two colors.
So lets prove by induction that a tree is bipartite.
Let be the proposition 'every tree with n vertices is bipartite.' The is true, which is trivial.
Suppose that and that is true. Let be a tree with vertices. Then contains a leaf . Let be be the vertex adjacent to and let be the tree formed by removing from . Then has a 2-coloring, and we can extend this to by coloring a different color to .
Thus every tree is 2-colorable.
As to the question about if all trees are planar, I wouldn't know what to do. What are your thoughts on this?