They used their spears while pushing. I also think that the formation was REALLY tight, because it is known that the shield of each soldier covered the left half of his body AND the right half of the body of the soldier right next to him.
It's not actually so clear, in fact. You should try adopting a hoplite fighting stance and see what it implies, considering the width of a hoplite's shield. You would be standing sideways-on and your right-hand side would actually be fully protected by your own shield - since after all it would pretty much be directly behind your left-hand side. I don't think the fact that the best soldier was - allegedly - placed on the right flank of the line actually says anything about the use of the hoplite shield. But at any rate, van Wees covers this in his book (I don't have it on me right now, unfortunately).

Consider though the fact that most Greek city states (except Sparta) did not actually have group formation drills, from what we can tell. That does not lead to such a close order phalanx, generally! Modern scholarship has moved on now from the view that the hoplite phalanx was some sort of close order 'battering ram', especially when you consider that the Spartans are recorded to have successfully held a defensive hoplite line against revolting helots that was only two ranks deep. And there's the fact that 'shoving' tactics would lead to the death by crushing of most of the front ranks.