Prove that spaces $C^k(\mathbb{R}^n)$ and $C^\infty(\mathbb{R}^n)$ are infinite dimensional. 
Prove that spaces  $C^k(\mathbb{R}^n)$ and $C^\infty(\mathbb{R}^n)$ are infinite dimensional. 

So in order to prove that they're infinite dimensional spaces, I need to form a linearly independent sequence which is infinite. Would I simply pick polynomials? (i.e. $1,x,x^2,\dots$ as my basis?)

Yes and no. Yes, because they form an infinite linearly independent family, and that is enough to prove what you want to prove. And no because they do not form a basis.
