Your first interaction term is bilinear in 2 gauge fields. Terms of this form indicate that you need to diagonalize your mass matrix. So you could work out the Feynman rules for your first interaction, but your question is sort of a moot point since you won't have to do perturbation theory for this term, it will get assimilated into the propagators in the diagonal basis.
EDIT:
Just so you get your orginial question answered, the Feynman rule for your first interaction is obtained by Fourier transforming the Action:
$S = \frac{g_1}{4}\int d^4 x F^{\mu} G_{\mu \nu}
= \frac{g_1}{4}\int d^4 x \left( \partial_\mu F_\nu -\partial_\nu F_\mu \right) \left( \partial^\mu G^\nu - \partial^\nu G^\mu \right)
$
$=\frac{g_1}{2}\int d^4 x \left( F_\nu \partial_\mu \partial^\nu G_\mu - F_\nu \Box G_\nu \right) = \int d^4 p \tilde{F}_{\nu} (-p) \left[ \frac{g_1}{2} \left( p^2 \eta^{\nu \mu} - p^\nu p^\mu \right) \right] \tilde{G}_\mu (p)
$
where the Feynman verte rule is in the square brackets now.
Now, to rephrase my original point, the Lagrangian
$\mathcal{L} = \frac{1}{4} F^{\mu \nu} F_{\mu \nu} + \frac{1}{4} G^{\mu \nu} G_{\mu \nu} +m^2 G^\mu G_\mu+ g_1 \frac{1}{4} F^{\mu \nu} G_{\mu \nu}$
is trivial, in that it is free. You can see this by doing perturbation theory, or you can make things easy on yourself and diagonalize the quadratic portion of the action.
Let me know if you want further clarification.
You can compute the feynman rule for the $\phi$-$\phi$-$\chi$ vertex by taking
$$e^{-i \int \mathrm d^4x L_\mathrm{full} }\frac{\delta}{\delta \phi^a} \frac{\delta}{\delta \phi^b} \frac{\delta}{\delta \chi^c} e^{i\int \mathrm d^4x L_\mathrm{full}}$$
where $L_\mathrm{full}$ is the sum of the free and interaction Lagrangeans and afterwards remove any propagator connecting to external degrees of freedom.
Best Answer
Try, for instance, section 9 of Srednicki. The way to do it is to replace the fields in the interaction Lagrangian by functional derivatives with respect to the sources, then write power series for the exponents. Take the first order contribution.
Then, use that you need to consider three-point functions where the fields are again replaced by functional derivatives.
Finally, you work out all these derivatives (either explicitly or making use of diagrammatic techniques) and see what numerical factor you end up with. Note that you'll have to use Grassman variables in your path integral because you're dealing with fermions.