New star formation will have died out eons earlier, so there will be no iridescent nebulae, supernova blasts, or blue giant stars like Rigel in the constellation Orion.
Our Milky Way will have lost its identity long ago through merging with the Andromeda galaxy, M31. The resulting giant elliptical galaxy will be devoid of dust and gas.
The night sky will be a largely homogeneous sprinkling of stars. Stellar density will concentrate toward the galactic core. There will be no bright arch of the Milky Way to obscure the view all the way into the core.
But as long as there are stars, there will be planets, and the possibility of intelligent life to gaze curiously upon the sky.
No Clues to Our Cosmic Roots
Ironically, astronomers living in that far future will have a cosmic view as simplistic as it was before giant telescopes discovered external galaxies, which in turn revealed the expansion of space. All far-future astronomy will be exclusively about characterizing stars, as it was in the 1800s.
A distant scientific civilization will have no clues about the birth and evolution of the universe. That is unless some grand "galactic archive" was set up to store information for far future generations. It would be the mother of all time capsules. Imagine very distant descendants trying to interpret the Hubble Ultra Deep Field - a snapshot of the universe in its heyday just a few billion years after creation.
By the year 1 trillion, the accelerating universe will have infinitely stretched the light from all external galaxies - assuming dark energy truly is Einstein's cosmological constant and not an unstable field that winds up destroying the universe. 100 billion galaxies will have winked out of sight long ago due to the ballooning of spacetime.
The plethora of white dwarfs, black holes, and neutron stars will be evidence that stellar evolution is a one-way street. Their existence will show that the universe cannot be eternal. This idea was embodied in the steady state theory that competed with alternative the big bang theory, until the cosmic microwave background was discovered in the 1960s.
However, the glow of the cosmic background from the Big Bang will be so weak as to be undetectable. There will be little evidence a Big Bang ever happened. However a far-future Einstein may hypothesize such an event based on the nucleosynthesis of elements in red dwarf stars.
The density of matter in space will be very dilute due to the dominance of dark energy, and difficult to measure. Future astronomers will conclude that matter must have been more tightly packed very long ago for stars to form through gravitational collapse.
Stellar ages and especially those of cooling white dwarfs will allow future astronomers to calculate when something happened in the universe to make so many stars at once. But they won't have a clue that our galaxy actually combined with another separate star city to trigger the explosion of star birth.