There are differences in acidity in different varieties of coffee, and you can choose a variety that is more acidic (more acidic coffees will often have their flavors described with terms that suggest brightness or fresh fruit). Roasting also breaks down some of the acidic components, so the darker the roast, the less acidic it will be. Try a light to medium roast for more acidity.
However, what you describe as related to brewing method probably isn't really tied to brewing method, but rather grind and extraction time. The acidic flavors are the easiest to dissolve, so they are the first flavors extracted, regardless of brewing method.
With espresso, you use a very fine grind and extract for a very short time. With French press, you use a coarse grind and extract for a long time. Given any combination of grind and water temperature, the flavors extracted will vary over time. The acidic flavors mostly extract relatively early. Then flavors that give it sweetness and body. The bitter flavors are the least soluable, so they are the ones extracted latest in the process. You can also minimize bitter flavors with low water temperature.
It's important to recognize that the apparent flavor is affected by the balance of the different flavor components. If you stop the extraction early, the acidic flavors will be over-represented. As the brew progresses, other flavors become more noticeable, so the acidic flavors are less noticeable in comparison. If the brew has plenty of body and flavor, and is noticeably acidic, that probably reflects the bean variety and roasting (or an excessive amount of coffee with a short extraction time). If the taste is thin and empty but noticeably acidic, that points to it being under-extracted.
With espresso, time and temperature are controlled, and grind needs to be within a narrow range for the process to work. With a French press, you can control all the variables. If you want a more acidic result, start with lower temperature water (try 175°-185°F), and brew for a shorter time. If the flavor is good but weak, use more coffee.