Visible vs Invisible Minority
A visible minority is easily distinguished from the majority by appearance. The most obvious examples are African-Americans and Asian-Americans in the USA or Arabs in Europe.
Invisible minorities are phenotypically indistinguishable from the general population. Chinese and Koreans in Japan, Jews in Europe and the USA look the same as the majority but differ culturally.
The social dynamic for visible and invisible minorities is very different. Visible minorities frequently face more open discrimination. Since they cannot 'pass' for the majority they may be unable to integrate and assimilate into the mainstream society and culture (due to discrimination from the majority). Their ability to attain a high social status can be also severely hindered.
Since the differences in appearance are easily perceived, much of the minority's identity is associated with looks. It is quite possible that culturally a visible minority is similar to the majority. For example, African-Americans in the USA share the same cultural values as whites. There are some differences, of course, but when it comes to core values and attitudes US minorities differ from other cultures more than from mainstream white culture. The perception from inside the minority, however, may not reflect this, especially when ethnic and racial tension is high.
Invisible minorities usually have cultures dramatically different from the mainstream culture. They can express it through outfits, occupations, and lifestyles. Religion is frequently the main reason for this. Invisible minorities have a lot of mechanisms preventing mixing with the majority. These can range from self-isolation to limitations in choosing marriage partners. The groups are tighter and probably have greater tendency to live together since it is important for maintaining culture.
With invisible minorities, there is always more intermixing with the majority. Since minority members can pass for a majority, some of them renounce their minority heritage and assimilate into the mainstream culture.
Stereotypes and Social Status
Both types of minorities trigger the creation of stereotypes and urban legends. Some of these stereotypes can be negative (think about black lazy criminals in the USA) and some can be positive (Asians as a model minority in the USA). Negative stereotypes are more prevalent, though, and positive are frequently accompanied by negative (in the USA Asian men are seen as smart but lacking sexual prowess).
It is possible that a minority has higher social status and/or higher income than the majority due to positive traits attributed to minority members or occupations that they traditionally choose. For example, in the USA, persons of Jewish ancestry are one of the highest earners despite their minority status. When it comes to race, Asians are the highest earners (same link).
Unfortunately, the majority of minorities have lower status and lower income due to a social stigma attached to them. It does not matter whether the minority is visible or invisible. For example, burakumin is a highly ostracised invisible minority in Japan. Burakumin are ethnically Japanese but during the Edo period (1603–1867) they became outcasts restricted to 'impure' occupations such as butchering and cleaning. Even today, despite all the anti-discrimination laws there are firms specialising in background checks of potential spouses to prevent marriages with people of burakumin ancestry.
Racial and Ethnic Tensions
It might seem as a good idea to depict minority-majority dynamic as a high-tension situation, especially if you or your audience are USA-based. However, American racial situation is rather unusual. There are many multi-ethnicity and multi-racial countries where it does not lead to constant social conflicts.
People who come to the USA from Europe are often surprised to discover how much race and ethnicity matter in America. In Europe the differences are handled in a much subtler way. The cultural emphasis also differs: one is a national of a particular country first and a member of a certain race or ethnicity next. In the USA it seems to be reversed: one's racial and ethnic status matters more than nationality.
How to Integrate It into Storytelling
I think that it is important to decide first whether your minority is visible or invisible. Then you can build on it.
If it is a visible minority, you can add scenes where someone reacts to their looks, makes a comment, or jumps to a wrong assumption. Depending on minority-majority relationships depictions of a riot or ethnic area might be appropriate. A couple of colourful details would bring things to life.
In case of invisible minority, I would focus on cultural differences and perceptions. Something like cultural misunderstanding can be an effective way to introduce your minority. Names can be also very important here. While mentioned-above burakumin are indistinguishable from Japanese in culture and appearances, their surnames betray their origins. Your minority characters may be forced to hide their names.
Of course, marriage is always a good way to highlight ethnical or racial tensions. Potential in-laws might be concerned with tainting their bloodline, losing social status, or cultural incompatibility. They can be supportive or not. There are plenty of variations here. And you can play with both families and their reactions.
I would try to be very careful to avoid stereotyping your minority characters. The differences among individuals within a group are usually higher than the differences between groups that live in the same area. I also think that it can make an interesting story if you show how an individual deals with their minority identity.