What if dark matter and dark energy don't exist and it's something else pulling (or pushing) the galaxies apart? What could it be?

They don't exist. Dark matter was invented to account for some difficult-to-detect matter that is inferred to be present due to the rotation curves of spiral galaxies and the lensing effects seen in distant galaxy clusters. The most sensible candidate for this missing matter is molecular hydrogen, the most abundant form of matter in the universe. Cold H2 is almost impossible to detect spectroscopically but we have found enough of it to account for the missing mass in a warmer galaxy than our own:

http://web.archive.org/web/20100702194353/http://h...

The reason that the scientific consensus says it can't be baryonic matter is that, if it was, it would give off thermal radiation of around 3K — the equilibrium temperature of interstellar space. Well, that's exactly what we see coming from all around us, the so-called "cosmic microwave background". The problem here is that the big bang theorists had already decided that this radiation is the left-overs of their hallowed creation event and they were not gonna admit that they'd got that wrong. So they invented this metaphysical non-baryonic [and non-emitting] stuff in a desperate attempt to keep the big bang theory alive and named it "dark matter".

As for dark energy pulling galaxies away from each other, that idea comes from the original misinterpretation of galactic redshifts as being a doppler effect when, in reality, they are a scattering effect. The galaxies are not generally receding from each other. Their light simply loses energy through it's interaction with the molecular hydrogen that fills intergalactic space (see source).

If there was any dark energy expanding space, as well as stretching out the energy of light radially causing distant galaxies to appear redshifted, it would stretch it out transversely causing a reduction in their surface brightnesses. This is not what we observe:

http://www.sci-news.com/astronomy/science-universe...

Expanding space would also stretch out the light curves of quasars (their oscillation in luminosity). No sign of this either:

http://phys.org/news190027752.html

This documentary may interest you:

https://www.youtube.com/watch?v=IFFl9S39CTM

There never was a Big Bang.

The whole idea of that is literally dark matter. No matter what it actually is, to us, it's dark matter.

"Dark energy" and "dark matter" are nothing more than placeholder terms for those unidentified "something elses". Once we identify them, we'll give them different names (or maybe not.)