Intuition behind Riemannian-metric

The idea is to equip the tangent space $T_p M$ at $p$ to the manifold $M$ with an inner product $\langle -, - \rangle$ , in such a way that these inner products vary smoothly as $p$ varies on $M$.

It is then possible to define the length of a curve segment on a $M$ and to define the distance between two points on $M$.

In terms of the definition of a Riemannian metric, given a smooth $n$-dimensional manifold, $M$, a Riemannian metric on $M$ (or $TM$) is a family, $(\langle -, - \rangle_p )_{p \,\in M}$, of inner products on each tangent space, $T_p M$, such that $\langle -, - \rangle$ depends smoothly on $p$.

I just came across this question as I was searching for answers in a similar direction and thought I'd share a couple of thoughts on this. My knowledge is very limited and I apologize for possible mistakes (in this case, I'd be very happy to be corrected!). In particular, my answers are certainly only partial and there are significantly many people here who can provide significantly more complete answers. Among others, you asked the following:

"we want to measure length of tangent vectors (but why?)": Think of basic calculus, where the length of a curve $\gamma: [0,1] \to \mathbb{R}^d$ is calculated as $\int_0^{1}|\gamma'(t)|dt$, provided $\gamma \in C^1([0,1],\mathbb{R}^d)$. We note that we need to calculate the (norm of the) velocity $\gamma'(t)$ at each time $t$ to obtain the length of the curve (intuitively: the velocity at time $t$ may be thought of as the length of the curve on a small time interval $[t,t+\epsilon]$, so we would try to sum the curve's length on all these time intervals. In the limit $\epsilon \to 0$, this summation becomes integration). Thus, analogously calculating the length of a (sufficiently smooth) curve $\gamma: [0,1] \to M$ in a smooth manifold $M$, we also need the derivative $\gamma'(t)$ at each $t$. However - what should the derivative of this $M$-valued path at any given time be? Well, the derivative is just the direction the curve passes through at this instant of time, i.e. at position $\gamma(t)$. The collection of all possible such directions is just the tangent space $T_{\gamma(t)}M$ at the point $\gamma(t)$. Hence, as in the example above, we need a means to measure the length of elements of each tangent space - for which the given metric provides a convenient tool. Then, the formula for the path lengths looks very much alike to the $\mathbb{R}^d$-case above. (Note: This general case is in accordance with the example above, as the manifold $\mathbb{R}^d$ has tangent space $\mathbb{R}^d$ at each point).

"Also, if we want to give a metric structure (to be able to talk about all things distance related) then why not define a metric (the topological one, satisfying positivity, symmetry, triangle inequality) rather than this?" First of all, note that we are NOT defining "a" (that is, one and only one) metric "on M": Instead, we introduce a metric $g_x$ on the tangent space $T_xM$ for each $x \in M$ (since in general, these tangent spaces need not coincide as sets, it is possibly not even meaningful to try to define the same metric, say $\bar{g}$, on each tangent space). We do so by introducing an even better object - namely a scalar product $g_x$ on each $T_xM$, which certainly induces a metric, but has a lot more convenient features (a metric only would not allow the length calculation as pointed out in the above answer, since this formula does require a norm). This induced metric is - put in your words - a "topological" one. Note, however, that this metric is NOT defined on the manifold itself: You do not measure distances of points on the manifold with this metric, but it is defined on the tangent space, which should carefully be separated.

"Or maybe I'm confused and Riemannian metric as we've defined is actually that metric(topological one) and together with it, our smooth manifolds becomes a metric space?" As said in the previous answer, the foremost goal of introducing a Riemannian metric is NOT to metrize the manifold M itself, but its tangent spaces! Please distinguish carefully.

Since you have posted your questions more than 18 months ago, this might be superfluous for you by now. But maybe others will come across this post and find it valuable :)