From my understanding the metric tensor is a function/tensor field that specifies a tensor at each point on a manifold, the tensor is used to define the inner product for the tangent space (in this sense the inner product can also be referred to as the dot product) and then from there we can naturally induce a norm and subsequently a metric for said tangent space at any given point on a manifold forming a metric vector space. Is this correct?
Strictly speaking, we should always refer to "metric tensor field" because it's actually a tensor field, although for convenience we simply refer to "metric tensor". You're right that the metric tensor field (usually denoted as $g$) assigns to each point $p$ of the manifold an inner product (which is of course a $(0,2)$ tensor) $g_p$ on the tangent space $T_pM$. After this, you're right: the inner product $g_p$ on $T_pM$ gives rise to a norm $\lVert \cdot\rVert_p$ on $T_pM$, which can in turn be used to define a distance function $d_p$ on $T_pM$. This is all just basic linear algebra/analysis/topology; the only difference being that now the vector spaces in question are the various tangent spaces to our manifold.
The only thing you should not confuse is the double usage of the word "metric". In geometry, the term "metric" refers to the metric tensor field $g$. The word "metric" in "metric spaces" is the distance function $d$. As long as you do not mistake one for the other (based on the way you wrote things it doesn't seem you're confused by this but I just wanted to emphasize it), your understanding is right.