What should I consider as an observer to measure the speed of cosmic objects? I mean for example if earth is the observer, then there might be entire galaxies travelling faster than the speed of light relative to earth. So according to Einstein relativity this shouldn't be possible so I want to know what I should consider as an observer  to measure cosmic objects' speeds.
 A: The laws of relativistic motion apply to "motion relative to local space" and that is what cannot exceed C.  If you want to measure the speed of an  object using a distant reference frame (i.e. distant galaxy measured from earth's frame) then there is no speed limit in the first place, and there are, in fact, galaxies that exceed C from our reference frame.
A: First about reference frames of value for cosmological observations. 
So, the preferred reference frame for cosmological measurements is the comoving one, meaning it is moving along with the average flow of galaxies due to the expansion. In that reference frame one is said to have zero peculiar velocity. We have to vectorially subtract our peculiar velocity. It is mostly due to the galaxy and local cluster moving wrt to the cosmic flow. The total is about 360 Kms/sec. When we do that adjustment vectorially right,, in the observations, we see the universe as on the average homogeneous and isotropic, and it is the reference frame where the CMB is also isotropic.
The comments and answer already explained that in fact galaxies can are are moving away from us faster than light. For galaxies at distances of about 14 billion light years, at a redshift of about z = 1.5, they are super-luminal (faster than light), but we still see them. We can see now out to the so called particle horizon, about 45 billion light years. 
The fact that they are faster than light does not break relativity. The galaxies are not moving that fast, just the space between us is expanding, GROWING, and at distances greater than about 14 billion light years the space is expanding faster than light. There is no problem with that, general relativity explains it and has a pretty good model of it, with many cosmological values measured. As @Cort Ammon said in his comment, this whole thing should conceptually bother you. Nevertheless it is true, and it is fun to read and understand it and try to get convinced it's real. See the wiki articles on the expansion of the universe, and maybe some of the math. Try https://en.m.wikipedia.org/wiki/Metric_expansion_of_space
For a good intro w/o the math. 
