Are there more transcendental numbers or irrational numbers that are not transcendental? This is not a question of counting (obviously), but more of a question of bigger vs. smaller infinities. I really don't know where to even start with this one whatsoever. Any help? Or is it unsolvable?

Hint: the set of algebraic numbers is Countable. 
For proving this, you can show that the polynomials with integer coefficients form a countable set, and that the set of their roots is also countable. The latter is just the Set of Algebraic Numbers (over $\mathbb{C}$). As the Algebraic Numbers over $\mathbb{R}$ form a subset of the former set, it must also be Countable. 
