In the image below what is meant by the items within the squares:
Box 1. Derivative f is the function from tangent plane p to to tangent plane f(p)?
Box 2. What is the meant by dot product derivative f at f(p) versus the dot of v,w at p? in particular the f(p) and p.
The notation $T_pM$ means "tangent space to the manifold $M$ at the point $p \in M$." Similarly, $T_{f(p)}N$ means "tangent space to the manifold $N$ at the point $f(p) \in N$."
If $f \colon M \to N$ is a smooth map between smooth manifolds, then its derivative at $p \in M$ is a linear map $df_p \colon T_pM \to T_{f(p)}N$.
A Riemannian metric on a manifold $M$ is a choice of (smoothly varying) inner product at each point of the manifold. In your situation, there are two Riemannian metrics in play: there is the metric $g$ on $M$, and also the metric $\overline{g}$ on $N$.
The notation $\langle df_p(v), df_p(w) \rangle_{f(p)}$ means the inner product of the tangent vectors $df_p(v), df_p(w) \in T_{f(p)}N$ with respect to the Riemannian metric $\overline{g}$ at the point $f(p) \in N$. That is, the notation is really $$\overline{g}_p(df_p(v), df_p(w)) = \langle df_p(v), df_p(w) \rangle_{f(p)}$$ and $$g_p(v,w) = \langle v, w \rangle_p.$$
Saying that $f \colon (M, g) \to (N, \overline{g})$ is a local isometry means both that (1) $\overline{g}_{f(p)}(df_p(v), df_p(w)) = g_p(v,w)$ for all $p \in M$ and $v,w \in T_pM$, and also (2) $f$ is a diffeomorphism.
Frankly, it will be very hard to learn any Riemannian geometry (as you seem to be doing) without a good understanding of smooth maps, their derivatives, and tangent spaces -- at the very least.