Raising your familiarity with the people from Hillsville who visit the Lost Valley will unlock requests that can earn you a variety of items. In past versions of Harvest Moon, you would give gifts to the villagers to raise your friendship, but in this version of the series you won't be doing that. There's really not that much you have to do at all!
Your friendship with a person is displayed as a person's Chemistry. You can learn what your Chemistry is with the visitors by looking in your Bookshelf, select the Encyclopedia, and then choose the Villagers menu. On each of the villagers that you've met, in the lower-right corner will display the person's Chemistry level with you.
When a villager reaches certain Chemistry levels, talking to the person may open up a request option. The villager will ask you to do something, usually a request to bring him or her something, and in exchange you'll receive a reward.
It's assumed that having a high Chemistry is important if you want to get married!
Increasing your Chemistry is pretty easy to do. You just have to interact with the person when you see him or her. Because of this, it is important to watch out for the villagers on your bottom touch screen. The villagers can pop up throughout the day, so slide the map around to look all over the valley. When you see a person, run over to say "Hi" to raise your friendship with him or her.
Villagers who are shopkeepers won't be buying or selling while they are wandering around in the valley area. These fellows will only be in shop-mode when they're in their correct areas such as in front of your house (Iris and Hanna), inside the animal barn (Hunter), or in Moon Valley (Doc and Brutus). Sam is always in front of your house.
You'll earn 0.5% by simply talking to a villager once per day. You won't see a half percentage appear in your Villager Encyclopedia, but just from talking it will take two days to earn a full 1%.
When it is a villager's birthday, you just have to talk to him or her to earn a boosted 3% Chemistry. Your night time dreams will remind you if someone's birthday is coming soon; that doesn't necessarily mean that the person will be around for you to give them a birthday greeting.
Completing a villager's personal request will earn between 3% and 4% Chemistry. These you'll get directly from the villager, versus requets you get as part of the game's storyline to return the four seasons to the valley.
For the Harvest Sprites, to raise their Chemistry you will have to ask them to work. While they're out in the field, talk to the sprite to increase friendship with him or her. The sprites move at the hour mark, so watch the lower map screen if it is almost X:00 to see where the sprites teleport to.