Gasoline is made of oil, so it sounds to a lot of people like if the United States produced more oil domestically that gasoline would get a lot chaper. But a new CBO report on gasoline prices contains this nice chart which shows that it's not so. Canada is a net oil exporter, Japan produces no oil, and the United States is a middle case. But it's Canada, not the US, that's in the middle case for retail gasoline prices. Why?
The issue is that oil is a globally traded commodity, so oil isn't really any more expensive in importing countries than in exporting countries. International price differences are driven by the fact that some countries have high taxes on gasoline, some (like the U.S.) have low ones, and others have subsidies. Many oil-producing countries have adopted misguided consumption subsidy schemes so it's empirically true that high-production countries tend to have low prices, but this is a coincidence not a strict causal relationship.
What increased oil production does do is alter a country's trade situation. Canada imports a lot of consumer durable goods, so the more oil they export to the United States the more KitchenAid stand mixers they can afford to import from Ohio. This can be a big deal (Argentina, for example, really needs to bolster domestic energy production to raise foreign currency reserves) but it's a different issue and it's not one the United States is facing.