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Calculating transition matrix elements can be difficult, and I have found myself needing to use hydrogenic electric dipole transition matrix elements a fair bit.
$$\mathbf{r}_{nlm}^{n'l'm'}=\langle\psi_{nlm}~|~\mathbf{r}~|~\psi_{n'l'm'}\rangle$$
Is there any online source that gives the nontrivial matrix elements above (for hydrogen obviously)?
As a partial answer, I took the advice of @JonCuster and pulled the following equations from "Quantum Mechanics of One- & Two- Electron Atoms" by Bethe and Salpeter. All results are in C.G.S units.
The sum of dipole transition intensities for initial state $(n,l,m)$ and final states $(n',l+1,m')$, where $m'$ runs over the three allowed values ($m'=m-1,m,m+1$), is given by
$$\sum_{m'}|\mathbf{r}_{n,l,m}^{n',l+1,m'}|^2=\frac{l+1}{2l+1}\left(R_{n,l}^{n',l+1}\right)^2$$
Similarly, we have for final states with $l'=l-1$,
$$\sum_{m'}|\mathbf{r}_{n,l,m}^{n',l-1,m'}|^2=\frac{l}{2l+1}\left(R_{n,l}^{n',l-1}\right)^2$$
Here, the symbol $R_{n,l}^{n',l'}$ represents the radial overlap between hydrogenic states $(n,l)$ and $(n',l')$. A consequence of the above equations is that the lifetime of a state does not depend on the magnetic quantum number $m$.
The closed-form expressions for the radial overlap matrix elements in their full generality are highly complicated. Quoted once again from Bethe and Salpeter (who themselves quote W. Gordon), for $n'\neq n$,
$$\begin{align*} R_{nl}^{n',l-1}&=\frac{(-1)^{n'-1}}{4(2l-1)}!\sqrt{\frac{(n+l)!(n'+l-1)!}{(n-l-1)!(n'-1)!}}\frac{(4nn')^{l+1}(n-n')^{n+n'-2l-2}}{(n+n')^{n+n'}}\times\\ &\times\left\{F\left(-n_r,-n_r',2l,-\frac{4nn'}{(n-n')^2}\right)-\left(\frac{n-n'}{n+n'}\right)^2 F\left(-n_r,-2,-n_r',2l,-\frac{4nn'}{(n-n')^2}\right)\right\} \end{align*}$$
where $F(\alpha,\beta,\gamma,x)$ is the hypergeometric function, $n_r=n-l-1$ and $n_r'=n'-l$ are the radial quantum numbers for the two states.
EDIT: I could almost swear that I stumbled upon a random website in which you could type in whatever transition matrix element you wanted and it would spit it out (presumably from a pre-calculated table). It had an enormous amount of dipole and quadrupole transition matrix elements (for sure electric, maybe magnetic?). I really don't think I was dreaming when I came across it, but now I'm not too sure since I can't find it anywhere. I think it may have been associated with the authors of this paper, but I'm not too sure.
[1] "Quantum Mechanics of One- & Two- Electron Atoms" by Bethe and Salpeter
[2] (German) W. Gordon: Ann. d. Pys. (5) 2, 1031 (1929) - radial overlap integrals are calculated here. They are of course expressed in units of the bohr radius.