The fact that the farther away we look the faster objects are moving only implies that farther objects are moving faster. That's how they got farther away. You can't infer if they are accelerating or decelerating from those statements.
To tell if some object is changing velocity, you look at it at two different times and see if it is going faster or slower, you don't look at some other object.
IF gravity was the only force acting on these objects, you would predict that the expansion of the universe would be decelerating, but it is not.
This is mostly correct.
First off, doppler shift in this context doesn't necessarily mean the objects are moving faster, just that the space between us and the object is getting larger. That may sound like the same thing, but there is a subtle difference.
If an object is actually moving away from us, yes, the light will be doppler shifted towards the red end of the spectrum in a classical way. However, there are some objects we can see that have a redshift which implies the source is moving away from us faster than light. This can't occur because of relativity so something else must be happening.
The answer is that space is expanding. When a photon of wavelength L travels through expanding space, the photon's wavelength is lengthened as the photon stretches. This is what produces most of the redshift we see.
Now, how do we tell that the universe's expansion is accelerating? Well we look for what are known as standard candles - objects (actually events) that have a standard luminosity no matter where they occur in the universe.
There is a special type of supernova (type 1a) that always has the same luminosity - it's a standard type of explosion that gives off the same amount of energy always. By looking at a range of these events over a range of distances, we can deduce that the rate of the expansion of the universe is accelerating by just plotting distances and redshifts.
If the universe's expansion is constant, there will be a linear relation between distance and velocity. If the expansion is slowing, the graph should curve away from this line in one direction. if the expansion is accelerating, the graph should curve away from the line in the opposite direction. When we perform this experiment, we find the graph curves away from the line in the direction that indicates acceleration.
This was a really freaking amazing result and was not at all expected.