Acceleration and Dark Energy Clarification

[something111]

Hi everyone.
I am having a few issues completely comprehending dark energy and the expansion of the universe, and I need some clarification on certain aspects of it.

Alright, so I believe that my issue lies within what really acceleration is.
I am aware than as an object's distance is increased from us, its speed increases proportionally.
However I've been told that the absolute total expansion rate of the universe has been declining ever since the big bang.
Using this model, I don't see where dark energy is necessary at all.
For the first matter, the reason why an objects recession velocity is increased proportional to distance is because there being more space in between the two objects, and therefore more space to expand.
For the second matter, the total rate is declining so I see no need for acceleration.
I know that I'm missing something and in no way am I purposing that the scientists today are wrong, I would just like to know what is wrong my my above reasoning.
Thanks in advance for the help; I'm a layman so please stick to light mathematical descriptions if that's possible! :)

 

[Chalnoth]

Yes, the overall expansion rate has been declining. However, in the last few billion years it has been declining slow enough that galaxies are now moving away from one another at an accelerated rate.

To see how this can occur, consider what would happen if the expansion rate is constant. The recession velocity of a far-away galaxy is given by $v = Hd$, where $H$ is the expansion rate and $d$ is the distance. If the expansion rate is a constant, then as the galaxy gets further away (as $d$ increases), the recession velocity also increases.