~ Modes
After you have a girl as your girlfriend, you will see her at the welcome screen when you turn on the game. She will wear random clothes and greet you according to the season or day of the week. (The background also changes every month.)
You will also notice that the save slot looks different. It will have your girlfriend's face when you click on it and the details on the left screen will have some more items. You can read more about the items' details here.
~ Real Time V.S. Skip Mode
After you start the game, you will see a screen with two options. One is labeled Real Time and the other is Skip. You have to chose one of them.
Note: I write this section from what I understand. Some information may be incorrect.
Real Time Mode
In this mode, the game's clock will match your Nintendo DS's clock. A day in your real life is a day in the game. You will start with limited "Activity Points" that will be consumed once you choose your daily activities. So you have to use your time and plan your daily activities wisely. You will see all different events such as birthday, Christmas, New Year's, Valentine's, school break, school trip, exam, etc. You will also have to use this mode in order to see the dreams in dream events.
Skip Mode
In Skip Mode, one day in the game is about a few minutes in your real life, just like when you play the first part of the game. You will be looping through a general week and there won't be any special real-life events (Christmas, birthday, etc.). Anyway, it is usually faster to try to unlock items or fulfill some requirements when you play this mode.
Switching between Modes
In both modes, you can date the girl and fulfill some of the game's requirements to unlock CGs and items. You can switch between modes by clicking the Option command and select "Mode Change". You can also save the game first and then start the game at different mode.
One thing to be careful of is that if you schedule a date with your girlfriend in one mode then you switch to another mode and schedule another date, it will mean that you don't go to the previous date and you leave your girlfriend waiting. (It will be shown as "デート すっぽかし" in the Calendar.)
What you do in one mode will be transferred to the other. The hearts you have gained, the locations you have unlocked, the message you have sent to your girlfriend, etc., will be carried over when you change mode.