Your context is a little off. Drilling by itself won't necessarily or drastically affect oil prices. The price of a barrel and the price at the pump are not directly and/or completely related.
But adding supply to offset demand will certainly drive prices down. Supply has to be supplemented with refinement capabilities and right now, America doesn't have the infrastructure to refine much more than we already do. We don't build refineries anymore.
Now, if there was a market to refine more oil, there would be a bigger push to build refineries. To do that, the cost of the assets would have to be offset by the potential profits. And what is the most expensive and restrictive part of "drilling for oil" or "refining" it? Land, legal and EPA standards, processes, procedures, red tape, etc.
We could bring a lot more oil into this country and drive prices way down in 5 years if we were as committed to it as we were to get a man on the moon.
Not if the supply carries higher costs. Forget the refineries; they are going to be seen as overbuilt in the next decade.