Besides growing plants and raising animals meeting people in Stardew Valley is a big part of the game (if you want it to be). So here is how to go about interacting with the townsfolk, I won't go into too much detail as I want you to experience things in game.
If you navigate to the heart tab in your menu, you'll notice a list of the people you are aware of. The little hearts by their names represents your relationship to them. Full hearts is good, empty hearts means needs work.
Meet people
Introduce yourself to the town folk by travelling around town and talking to the people you meet.
Meet townsfolk
Friend level
Tips
Talk to everyone you meet. This not only learns you their name and story, but is also how to complete an early quest.
Give people gifts as this will make them like you more. You can give people two gifts a week. To learn what they like you can talk to them or the people around them, other than that you can try trial and error.
The towns people have birthdays, a calendar showing which day the characters birthday is appears on the front of Pierre's general store. Giving charters a presents on their birthday will make them like you more.
Being friends with the townsfolks will unlock dialogue and quests, as well enabling you to get cooking recipes and gifts in return.
What do you want to do now?