When the game first begins, the Church is not open. The village doesn't have a minister to run the Church. On the 15th of Spring, first year, Cain will arrive at the village to open the Church up.
There isn't much to the Church. When you enter it Cain will ask if you would like to pray to the Harvest Goddess. After you've finished praying, you can talk to Cain by selecting the "talk" option. The only other option available is "exit". Like I said, there's not much to the Church.
No one knows yet what the benefits of praying to the Goddess are. In the first HM game if you prayed enough times, one of your animals would be free from sickness forever. However in HM2, praying doesn't appear to have any affect on anything. The benefits of the Church will be unknown until Natsume or VIS lets us know.
There is one event that is held by the Church though. On the 25th of Winter every year the Holy Night Festival takes place. If you pray to the Goddess that night, says Cain, she is sure to appear. You are given 5 selections to pray for - Health, Wealth, Sunshine, Happiness, and Friendship. Don't bother to pray for wealth. The Goddess won't be too pleased with you (greedy farmer!). Sunshine will result in a sunny day the next day, and health makes you healty. Naturally you're already healthy though, so both of those are pretty pointless. The only two left that could be of any use is happiness and friendship. If you pray for happiness I would assume that your happiness rating goes up (need a high happiness to save the village at the end of 3 years) but how much it goes up is unknown. If you pray for friendship the Goddess says that she deepend your friendship, but she doesn't say with who.
After the 3 year mark there will appear something new in the Church. The music/sound test will arrive for you to use. You can scroll through the game's music and listen to the sound affects. There are 19 music tracks and 33 sound effects. Below are midi files of the 19 songs, recorded using Virtual Gameboy.