You'll first want to check either the discriminant or the eccentricity of your conic before proceeding to use any expression(s) for the foci; the central conics have two foci while the parabola only has one.
For the central conics, it is known that the two foci are at a distance $a\epsilon$ from the center, where $\epsilon$ is the eccentricity and $a$ is the semimajor axis for an ellipse, and the semitransverse axis for a hyperbola.
To simplify things, you'd first want to perform a translation on your central conic, such that the conic's center is now at the origin, and the standard equation becomes
$$\alpha x^2+\beta xy+\gamma y^2=1$$
with that, you can use the formula
along with the eccentricity formula (like the one here) and the formula for the slope of the major/transverse axis to figure out the coordinates of your foci.
The parabolic case is a bit tricky, and I'll leave that to another answerer.