What are galactic speeds measured against? The Earth moves through space at 67,000 MPH. The Milky Way travels through a local group at 2,237,000 MPH.
Wouldn't you need a fixed point to be able to measure velocity against? After all, compared to the total speed of our Milky Way, the Earth isn't moving through space. What fixed points do we compare against? When we say the speed of light is $c$, what is that relative to?
 A: There are two separate questions there. The easiest one to answer is how we measure the velocity of the Earth, Milky Way etc, because we measure it relative to the cosmic microwave background (or CMB).
If you measure the CMB in all directions and find it's the same in all directions then you are stationary in comoving coordinates. However if you find the CMB is blue shifted in one direction and red shifted in the opposite direction then you know you're moving relative to the comoving frame, and the change in the CMB is due to the Doppler shift. From the size of this change you can calculate your velocity. Measurements of the CMB from the Earth show exactly this Doppler shift, and that's how we can work out the velocity of the Earth. Having got this we can convert velocities measured relative to the Earth into velocities measured relative to the comoving frame.
There are traps for the unwary here, because all velocities are relative and the comoving frame is in no sense an absolute way to measure velocity. It just happens to be a useful reference and one that tallies what our instinctive interpretation of velocity relative to the rest of the universe. This is discussed in the question Assuming that the Cosmological Principle is correct, does this imply that the universe possess an empirically privileged reference frame?.
Lastly back to light. The velocity of light is special because every observer who makes a local measurement of the speed of light will always get the same value of $c$, regardless of what their velocity is. This is one of the building blocks of special relativity.
A: Speed is a distance (separation between two well defined points in space) traveled over a time. The speed of Earth you quote is the orbital velocity. We know how far away the Sun is and we know the shape of the orbit, so we know how far the Earth travels relative to the Sun (distance) per year (time). Likewise the speed of the Milky Way is also given relative to another galaxy or constellation. The answer to the broader question is that the speed of any object is relative and always given with respect to another object or reference frame.
The speed of light is likewise measured as the distance traveled by light over the time it took the light to travel that distance. Interestingly, no matter the separation of the start and the end point, the speed of light in a vacuum is always the same. 
A: There are a number of different frames of references.
For the velocities of celestial objects we use: (i) The geocentric frame: This is a velocity measured with respect to the Earth's centre. Obviously this is quite useful for artificial satellites, but also for things like meteors. (ii) The heliocentric frame: this is the velocity as seen from the centre of mass of the Sun. Heliocentric corrections to measured velocities from the Earth correct for the Earth's rotation and the motion of the Earth relative to the Sun (corrections of order tens of km/s). This is often the frame used for stellar velocities. (iii) The barycentric frame: This is similar to the heliocentric frame, but now referred to the centre of mass of the solar system. The difference between the two is only of order 10m/s, but this is important when discussing the velocities of stars when looking for doppler shifts due to exoplanets. Also crucial when looking at timing analysis from pulsars. (iv) The local standard of rest: this is set to follow the mean motion of objects in the vicinity of the Sun. The Sun actually moves at about 20 km/s with respect to the defined LSR. This frame is often used for discussing the motions of objects in our Galaxy (orbits of stars around the centre etc.).
The speeds of extragalactic objects are usually heliocentric velocities - precision is not usually an issue. However if one wished to convert heliocentric/barycentric velocities to the frame of rest of the cosmic microwave background, then the solar-system baycentre moves at $368\pm 2$ km/s in the direction $l=263.85 \pm 0.10$, $b=48.25\pm 0.04$, where $l,b$ are the Galactic latitude and longitude in degrees.
Your last sentence is answered in the comments. The speed of light is a defined quantity and is the same in all frames of reference.
