Well, that depends on what kind of problem you're setting up. From your wording, it sounds like this is a problem on Newtonian gravity, and you're ignoring things like dark matter and dark energy. In that case, it's just a matter of using Newton's law of gravity: the force between two objects with masses m₁ and m₂ is
F = G*m₁*m₂/r²,
where r is the distance between them. From the force, you can calculate the acceleration (and since gravity's attractive, you know that the acceleration of each galaxy points towards the other one).

Of course, in the actual universe, things can get a lot more complicated. As you may know, the universe started with the Big Bang, and has been expanding ever since. In fact, the rate of expansion of the universe is increasing, so the universe is expanding faster and faster as time goes on (we call whatever is causing this expansion "dark energy"). That means that without the presence of external forces, everything in the universe is accelerating away from everything else. In the case of, say, two galaxies, there's the gravitational attraction that tries to make the galaxies move closer to each other, and the expansion of the universe that tries to pull them apart. To figure out what their relative acceleration is, you'd need to know how strong the dark energy is that's pulling them apart, and how strongly gravity's trying to pull them together. This is very complicated to calculate; in practice, this question is generally answered by the observations of astronomers.

Well, if in your hypothetical universe only two same mass galaxies exist, then the acceleration of galaxy A towards galaxy B is given by
F/MA= G * MB/R²
where R is the distance from galaxy A to galaxy B and MB is mass of galaxy B and MA is mass of galaxy A.

Since they are the same mass, galaxy B is accelerated towards galaxy A as well. This is Newton’s third law.

Hmm, well, you need to know a couple of other things. First off, you would need to know the distance between the two galaxies, and second, you would need to know the value of what I'm going to call the "cosmological repulsive force" over that distance.

The first of these, the distance, is straightforward: using Newton's theory of gravity, which is perfectly accurate for calculating the gravitational attraction of two galaxies, the force between them would be equal to the product of their masses (thus, about 2 x 10⁸⁴ kg²) divided by the square of the distance between them. You need that distance.

The second property, the cosmological repulsive force, is harder to quantify, because it is not currently understood physically. Albert Einstein's field equations that describe gravity under general relativity include the possibility of a cosmological constant, which would be a universal, repulsive, gravitational force that causes space itself to expand a constant rate, which would manifest on matter in space as an acceleration that scales linearly with distance. Most physicists, including Einstein himself, considered this an inelegant explanation for the universe, and dropped it immediately when Hubble discovered that the universe is expanding, predicting instead that the universe is merely coasting on its outward momentum from an explosive event (the Big Bang) in the distant past. However, more recent observations of the speed of galaxies moving away from us indicate that the universe is not only expanding, but that the rate of expansion is *accelerating*, indicating that there *is* some kind of repulsive gravitational interaction that scales with distance, much like Einstein's cosmological constant. The most popular theory right now is that there is a form of vacuum energy known as "dark energy" or "quintessence" that exerts this repulsive force, but what the nature of this dark energy is, why it exists, and what its properties are is still a matter of speculation. The amount of it in the universe may or may not be constant - although we do, of course, know that the universe must have accelerated violently during the Big Bang itself, and is not accelerating at nearly that rate now (if it were, the repulsive interaction would be strong enough to tear atoms apart, and matter as we know it would not exist).

The expansion rate of the universe itself appears to be broadly constant, scaling linearly with distance as predicted by Einstein's theories, but is slowly accelerating. I don't know the rate of acceleration, but the current expansion rate, known as the Hubble constant, is approximately 70 km/second * megaparsec, that is, for every megaparsec two galaxies are apart (about 4.8 million light years), the space containing those two galaxies will be moving apart at 70 km/second. So, for example, if two galaxies are ten magaparsecs apart, then they will be moving apart at 700 km/second, assuming no momentum or other forces relative to each- other.

It should also be pointed out that the expansion of the universe is a product of space itself, not the movement of matter within space. Matter and energy within space cannot move faster than light, but space can do whatever it wants. This means that, beyond about sixteen billion light years' distance, two galaxies will be moving apart faster than the speed of light. This also means that light emitted by one galaxy will never reach the other - a cosmological event horizon exists between them.