There are five different girls in the game that you can develop a relationship with, so long as you talk to them, and give them the right kind of stuff. I think they all pretty much like flowers (Popuri more so), food, and whatever else you grow or find in the mountain.

It takes longer to get the girls to like you in Harvest Moon 64 (compared to the SNES original). You can't read their diaries like before, so you can tell how much they like you by the color of the heart at the end of their text-box. It starts out white, and if you give them enough gifts, talk to them, meet them at festivals, etc..., it will eventually change to a blue heart. Depending on how well you treat them, it should be possible to get a blue heart by the end of Spring, although sometime in the beginning to mid-summer is more likely.

The order of hearts, from the indifferent beginning to true-love is:

Once you get a girl to like you, they will start coming by your farm to give you gifts. They tend to come by in the afternoon, but not every day. You can only talk to them when they come over, so be sure to come over earlier (or later, depending on the time) so that you can give them gifts.

If you want to get married later in the game, you'll have to work your way to a red heart.

To get a more detailed version of each girl and what they like, go to their individual character pages.