To view your Chemistry with someone go check your bookshelf, select the encyclopedia, and then choose the Villagers menu.
Since there is no gifting in this game so to increase chemistry with someone, you have to talk to villagers to earn 0.5% once a day. When it is someone's birthday you will earn 3% with them when you talk to them that day.
Requests are another good way to increase chemistry. When fulfilling requests you earn between 3% and 4%
For Harvest Sprites, you will first have to ask them do some work and while they're out working just simply talk to them.