What is the effect of this expanding universe on the energy density of it? We know that our universe is expanding i.e.  volume of the universe is getting larger with time with total energy being conserved. My question is whether the energy density is getting smaller with time.
If this is true then what is the effect of this phenomenon on all the celestial bodies?
I am no expert and I am sure there will be a more rigorous answer later, but here my 2 cents:

It appears to me that the term "energy" and energy conservation becomes a bit more complicated in general relativity and cosmology, taking into account more abstract forms as the energy of the gravitational field or space curvature. As you seem to be more interested in the implications I just want to focus on the common forms of energy, i.e., radiation, matter and dark energy. 

For a given expanding volume of space only the energy from matter (dark and normal) is dilutes as expected. You could say that with respect to expansion its energy is conserved. This means its energy density decreases with $\sim a(t)^{-3}$, where a is the scale factor (or just the diameter) of the universe. 

Radiation additionally looses energy via cosmological red-shift, so its energy density decreases even faster with  $\sim a(t)^{-4}$.

Finally you have the dark energy density. What makes it special is, that its density does not depend on $a(t)$. One can think of it as the energy associated with space itself. So if the universe expands new space and dark energy "appears" and its density stays constant.

So over time the average (matter and radiation) energy density decrease and approaches the energy of empty space itself (or vacuum energy). 

Note, however, that this is only true for the average densities. In regions of space, that are held together by gravity, as our local group, the matter density will not decrease. On the contrary, these region are still contraction due to gravity. 

So to answer your question on the effect on all the celestial bodies:
Those which are gravitationally bound, won't notice much but others will drift apart never to see each other again.