What is the standard reference point for measuring speed?
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?
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.