The more distant the galaxy we observe, the faster it recedes from us. But the evidence that the expansion of the universe is accelerating is not due to the fact that the further we look, the faster the objects are moving away, which would be the case with any kind of expansion (decelerating, constant or accelerating).
Imagine a rubber band, with equal distances marked off on it at intervals of 1cm. When we stretch the rubber band, all the distance markers separate from their neighbouring markers at the same rate. All distances across that rubber band increase by the same factor, all the markers separate equally as it stretches.
Hold an end in each hand and stretch the band until all the markers are twice as far apart as they were to begin with. Now look at the viewpoint from one of your hands. The marker right next to your hand is now twice the distance away that it was before you started stretching it. But if you look at the tenth marker along, you will find it is now twice the distance away it was to begin with too.
Let's say you took 1 second to stretch the band to twice its original size, making all the markers twice the distance apart that they were to begin with.
The 1cm mark is now 2cm away, and the 10cm mark is now 20cm away. The 10cm mark has travelled 10 times the distance from your hand that the 1cm mark did, in the same amount of time. The first mark moved at 1cm per second, whilst the 10th mark moved at 10cms per second.
The further away a marker was from your hand, the faster it moved away from that hand, and this is what we mean when we say the further away an object is, the faster it recedes.
At any given time across that rubber band, distances are increasing at the same rate as each other, so all markers separate equally.
It doesn't matter if you start stretching the band fast and then slow it down, or start stretching it slow and go faster and faster - the furthest point you measure will always be receding the fastest.
So, the observation that, the further away a galaxy is, the faster it is apparently receding, is not what tells us that the rate of expansion is accelerating. All it tells us is that the universe is expanding metrically, like the rubber band.
What tells us that the rate of expansion is accelerating are observations of a certain type of supernova (known as "standard candles" because of their predictable properties), across a range of distances. We know that the rate of expansion was decelerating for billions of years and had expected this to still be the case, but we found that the closer the supernovae we looked at, the dimmer they were than is to be expected in a decelerating universe. The closer we looked, the further away those supernovae were from the predicted distance. As we look at them at increasing distances, the closer they get to the predicted value for deceleration.
The conclusion is that, around 5 billion years ago, the rate of expansion ceased in its deceleration and started to accelerate.
The rubber band was stretching incredibly fast to begin with but immediately slowed and continued slowing for around 8 or 9 billion years. Then it gradually started to get faster again. So, when we look at the closest markers, they are further apart than they would have been if the stretching of the band had continued to slow.