The times are "sunrise" "daytime" "sunset begins" (can be called evening) "sun sets". When the time changes one of those messages will pop up in the middle of the screen, and the color of the screen will change. Evening has a red hue over everything.
If it's raining, you can always eat a meal at the bar to pass time, or depending on what time of day, you can still find work. If it's sunrise time, you can talk with either Fina or Alice, if it's daytime then you can talk with either Astra, or Eric. Evening you'd need to dance pass the time in the bar. You can also always go to sleep early, to start a new day.
It's been awhile since I've done the tutorial, but I think the first objective you're shown is to get enough gold for the magic book in the shop, which is 50 gold.