How are the distances to galaxies measured?

By a variety of methods. The most direct one is the Cepheid Variable technique.

Spot a Cepheid variable star in a nearby galaxy, measure its period, look up on a calibrated graph its luminosity from the period you got, then use the star's apparent brightness with proper extinction corrections added, to get its distance. This works for the Hubble Space Telescope out to the Virgo Cluster.

For more distant galaxies, you can use Supernovae of Type 1B determined from their spectra and 'light curves'.

At the most extreme 'cosmological' distances of over 1 billion light years we use the recession speed 'redshift' and the Hubble Law you get its distance. This relationship has to be independently calibrated for it to be reliable, but currently it works to about 10% accuracy.

There are also many other indirect techniques which form the Cosmological Distance Ladder.