Theorem 1:
If $f : [a, b] \to [c, d]$ is a $C^1$ function and $g : [c, d] \to \mathbb{R}$ is continuous, then $$\int_{a}^{b} g(f(x))f'(x) dx = \int_{f(a)}^{f(b)} g(y) dy.$$
I want to prove this theorem directly from the definition of Riemann integral.
If $f$ is a monotonically increasing or decreasing function on $[a, b]$, then I guess the proof of Theorem 1 is easy.
Are there real numbers $x_0, \cdots, x_n$ such that $a = x_0 < x_1 < \cdots < x_n = b$ and $f$ is increasing or decreasing on $[x_{i-1}, x_i]$ for any $i \in \{1, \cdots, n\}$ if $f$ is $C^1$?
If there exist such real numbers $x_0, \cdots, x_n$, I guess the proof of Theorem 1 is easy.
Proof using the fundamental theorem of calculus and no assumption of monotonicity for $f$
Define $\displaystyle G(x) = \int_{f(a)}^{f(x)} g(t)\,dt$. Since $g$ is continuous, by the FTC we have
$$G'(x) =g(f(x))f'(x)$$
Since $G(a) = 0$ and $G'$ is integrable it follows that
$$\int_{f(a)}^{f(b)} g(x)\,dx = G(b) = G(b)-G(a) = \int_a^bG'(x)\,dx = \int_a^bg(f(x))f'(x)\,dx$$
Proof using the "definition of Riemann integral" when $f$ is increasing.
It seems you are looking for a proof using Riemann sums when $f$ is increasing. I provide one here. This uses the mean value theorem and the fact that both $a < x_0 < x_1 < \ldots < x_n = b$ and $f(a) = f(x_0) < f(x_1) < \ldots < f(x_n) = f(b)$ define partitions. Also, in this case, it is only required that $g$ is Riemann integrable.
The most general theorem, of which I am aware, does not require continuity of $g$ or $f'$, but is much more difficult to prove.