So I guess he steers in the old fashioned way and all the button pressing would be him getting the right read outs on the screens to he know were he is and whats around him...
Here's another view, from what is clearly an on-set photo from the original trilogy (Luke has his hand cut off, so this is from Empire):
Notice Leia's hand is resting on some crontol that is approximately where the control yokes pictured in the top image would be. Also, we can see the closer control yoke and the handgrip pointed to by the arrow is clearly the same shape as the handgrips visible in the top photo. So even though the provenance of the top photo is unknown, it matches closely enough to what is in the bottom photo, which is clearly from the original trilogy and a shooting set, that I think we can consider the top photo to be authoritative.
So The Millenium Falcon is most likely controlled in a way very similar to a typical modern Earth aircraft. Note that we also know that X-wings have control sticks, much like modern Earth fighter jets, so it's reasonable to assume that they are controlled in a similar manner.