When two people go out together because they are or want to be in a romantic relationship, this is often called dating. When someone goes on a date, they try to do things that both people will like doing: go to see a movie, eat in a restaurant, drink in a pub, bar or café.
Dating can be done with both new people and people one already is in a relationship with. A first date lets people decide whether they are interested in that person, while dates after that are done to try and build the relationship.
There are commercial services to help people find people that they might want to go on a date with: dating services or introduction services. On the Internet, there are dating sites that allow people to find people who they might want to date.