The race starts on 10th day of the game, if you entered it earlier that's probably a bug. There's no way you could complete all quests in time, you should choose two or three girls you like and focus on them. Then, after the race you choose only one of this girls to become closer with her. To see all content in the game you will need to play it multiple times. The game consists of 3 chapters, each chapter lasts 10 days (one month in total) and only half of it available for now. Also I think it's abandoned or on break since the last REAL update was more than a year ago.