UserLTK's answer is correct but Id like to expand on it. It is not simply expanding out and decelerating or accelerating. The speed at which galaxies are flying away from each other increases as their distance increases, and that is an important and not obvious factor.
We know this because as the galaxies get further away from us their cosmological redshift increases, meaning they are going faster.
This has to do with the fact that the spacetime is really what is expanding. What the Big Bang did was to make spacetime grow, and it grows at an average rate, now, of 70 Kms per second, for distances of 1 megaparsec (about 3 million light years), and faster at larger distances so for galaxies about 100 megaparsecs away they are receding from us on the average at 7000 Kms/sec.
One way to visualize it is to consider the surface of a balloon as you inflate it, and mark 3 points on it, two a distance of 1 inch away, and the third one 100 inches away measuring along the spherical surface. That surface is like our 3 dimensional universe, which is expanding. The point 100 inches away goes away faster from the other two, than those two go away from each other. Our universe is like that surface (except the universe at any one instant is like a flat sheet and you are stretching it, the same idea). The points you drew were like 3 galaxies.
And the universe has been expanding since the Big Bang.
So, that expansion has been happening, and the expansion parameter is the 70 km/sec per megaparsec. That is the Hubble parameter, now. In the past it was different. The question is how has it changed, and will it change in the future. The Hubble parameter was indeed higher at earlier times, but over the past 6 billion years or so it's been getting larger. Ie, for the last 6 billion years the dark energy, which contributes to the universe mass energy has been dominating more and more (still a small effect, it changes slowly), and we are now accelerating. We decelerated for the first few billion years. The dark energy causes acceleration because it has a negative pressure that counteracts its mass density, and causes a accelerated expansion instead of a slowdown.
Still ,even now there are galaxies that are going away from us at the speed of light and faster. Those that are about 10 billion light years away (and unfortunately I forget the distance at which this happens, but it is within the observable universe of about 13.8 billion light years) are NOW receding from us faster than light. We see their light because it was emitted before, about 10 billion years ago, and they were not going that fast then.
So, even apart from the acceleration, it is still 'strange' thatgalaxies are moving away faster the further they are. That is because of the expansion of the universe, those further away go faster.