Speed, as we know, doesn't exist without first having a reference point. We then say that the reference point isn't moving at all, and speed is then measured in relation to the reference point.
What is the standard reference point when determining the speed of objects in astronomy? Earth?
As Einstein realized and like you correctly state, you indeed can't measure the speed of an object by itself since it has to be measured relative to something else.
As a logic result, if you ask the question "How fast is X moving?", you will have to specify that you want the speed with respect to another object because motion cannot be measured without a reference point.
Some examples:
"standard" reference point
If no reference point is given, it could be assumed that the "standard reference point" is the location of the observer. In the realms of astronomy, that would be the location of the astronomer.
It should be noted that the astronomer could well be on an space mission outside Earth's atmosphere, or he/she could be using a telescope in Earth's orbit. Therefore, "Earth" can not generally be assumed to be a standard reference point for astronomy because, depending on the precision of the measurements and the location of (for example) the telescope, assuming "Earth" may result in inexact measurements.
EDIT
As @TidalWave correctly commented, there's also the International Celestial Reference Frame (ICRS) which can help you find reference points, calculate distances, etc. according to a celestial reference system, which has been adopted by the International Astronomical Union (IAU) for high-precision positional astronomy. The origin of ICRS resides at the barycenter of the solar system.
Wrapping it up:
If no reference point is defined and if we can assume that the astronomer works according to the rules and definitions of the International Astronomical Union (which would be the regular, if not ideal case), the International Celestial Reference Frame provides you with a "standard reference point" at it's center (which is the barycenter of the solar system).
In rare cases where IAU compliance can not be assumed (something that "might" happen in amateur realms), it has to be assumed that the reference point is the point of observation.
The rest frame for measuring (astronomical) speeds and velocities depends on the context and purpose.
A geocentric frame, based on the Earth's centre of mass might be appropriate for objects in orbit around the Earth.
A heliocentric reference frame, centred on the Sun, is often used when describing the line of sight velocities of astronomical objects, but more usually, a barycentric frame at the solar system centre of mass is used. For example, this is how the radial velocities measurements of exoplanet host stars or components of eclipsing binaries would be quoted. It would also be appropriate for the motion of objects in the solar system.
Motions in the Galaxy are commonly defined in two ways. One is the Local Standard of Rest (LSR), which measures velocities with respect to the average motion of stars near the Sun. The second attempts to measure velocities with respect to the Galactic plane and radially and tangentially in the plane with respect to the Galactic centre. These are the so-called $U,V,W$ velocities. The position and motion of the LSR with respect to the Galactic centre are uncertain at the level of a few km/s, whereas the solar motion with respect to the LSR is more precisely known.
The Galactic centre can also be used as a reference frame for the motion of galaxies in our local group. e.g. when talking about the motion of the Andromeda Galaxy.
Finally, we can determine a cosmic standard of rest - the co-moving stationary frame of the Hubble flow - using precise measurements of the cosmic microwave background (CMB). In other words, we can determine our peculiar motion with respect to the Hubble flow by observing the dipole doppler signature in the CMB.
