Gas prices are a simple function of supply and demand. As it stands, the middle east has a virtual stranglehold on oil, which leads to all sorts of sticky political situations. If we drilled more domestically we could be oil independent and the increase in supply would necessarily push down gas prices. It's just that simple.
Drilling on domestic land for oil resources is a tactical decision, and not a very environmentally friendly one at that. Drilling locally can help bolster fuel reserves while we explore other energy options. It would lower gas prices and relief some pain at the pump, but the dependence on fossil fuel won't be resolved by this step alone.
There is already enough oil in North Dakota to fuel the United States for the next 2000 years. All that is needed is the permission to mine it. I don't care how greedy you think the oil companies are, if they have a surplus, the price would have to go down. It's the simple concept of supply and demand. While it may cost more to mine the oil in the US, it costs far less than it would to ship it from the Middle East and our domestic situation would be a lot safer. Prices would have to go down. It's common sense.
If we drilled domestically the price would go down because it would decrease the reliance on Middle Eastern countries. It would increase the amount of oil available and since supply would increase, demand would then decrease. The oil wouldn't have to be transported over seas so the price of moving the gas wouldn't be factored into the gas price so it would also decrease the price.
It's never really been about supply. Of course importing oil from the Middle East will be more expensive than domestic, but the price of oil tends to vary somewhat randomly, so it's far from a rock-solid relationship. Every time gas prices soar, oil companies post record profits, so they're certainly not going to give up these profits simply because of a different supply.