Basic Mathematical Induction I'm not quite sure how to approach this question.
I need to prove that for $$n\ge1$$
$$1^2+2^2+3^3+\dots+n^2=\frac16n(n+1)(2n+1)$$
Do I just plug $1$ and see if $$\frac16(1)((1)+1)(2(1)+1) = 1^2\text{ ?}$$

No, that is only the first step (the base case.)  Not every property of the number 1 holds for all positive integers.  So next you need to verify the induction step. Namely, assume that we have a positive integer $n$ with the property that
$$1^2+2^2+3^3+\cdots+n^2=\frac16n(n+1)(2n+1),$$
and prove that $n+1$ has this property also:
$$1^2+2^2+3^3+\cdots+(n+1)^2=\frac16(n+1)((n+1)+1)(2(n+1)+1).$$
