Let V be a $\mathbb{R}$-Vector space with inner product $\langle •,• \rangle : V\times V \to \mathbb{R}$ Notice $||v||=\sqrt{\langle v,v \rangle}$ Define a norm on V$$
What are they specifically asking me to do? Do they want me to show that $||v||=\sqrt{\langle v,v \rangle}$ is a norm by showing it satisfies homogeneity, non-negativity, Triangle inequality and that $\|v\|=0$ if and if only $v=0$, or do they want me to do show something else?
Yes. You have to check that that function $\|\cdot\|\colon V\to \mathbb{R}$ given by $$ \|v\|=\sqrt{\langle v,v\rangle} $$
satisfies the axioms of the norm that you mentioned.