Taylor coefficients and series involving harmonic numbers

During 2022–2023 Z.-W. Sun posed many conjectures on infinite series with summands involving generalized harmonic numbers. Motivated by this, we deduce $58$ series identities involving harmonic numbers, eight of which were previously conjectured by the second author. For example, we obtain that $$\sum _{k=1}^{\mathrm{\infty }}\frac{(-1{)}^k}{k^2\left(\genfrac{}{}{0px}{}{2k}{k}\right)\left(\genfrac{}{}{0px}{}{3k}{k}\right)}(\frac{7k-2}{2k-1}{H}_{k-1}^{(2)}-\frac{3}{4k^2})=\frac{{\pi }^4}{720}\mathrm{.}$$ and $$\sum _{k=1}^{\mathrm{\infty }}\frac{1}{k^2{\left(\genfrac{}{}{0px}{}{2k}{k}\right)}^2}(\frac{30k-11}{k(2k-1)}(H_{2k-1}^{(3)}+2H_{k-1}^{(3)})+\frac{27}{8k^4})=4\zeta (3{)}^2,$$ where $H_n^{(m)}$ denotes ${\sum }_{0<j\le n}{j}^{-m}$.