I am new to programming in general so please keep that in mind when you answer my question.
I have a program that takes a large 3D array (1 billion elements) and sums up
Though this would probably be very challenging to you if you're new to programming, a very powerful way to speed things up would be to use the power of GPU. Not only is the VRAM much faster than usual RAM, the GPU can also run your code in parallel on some 128 or more cores. Of course, for this amount of data you will need to have a pretty large VRAM.
If you decide to check this possibility out, you should look up nVidia CUDA. I haven't checked it out myself, but it's meant for problems like this.