Boutiques - Every day the items in the boutiques changes. These are random changes so you have to visit them frequently to find something you might want.
Hotels - NPCs moved around to different hotels each day. But their is a lady that gives you a ribbon on different days.
Saturday - Cyllage City - Snooze Ribbon
Sunday - Courmarine City - Smile Ribbon
Monday - Ambrette Town - Alert Ribbon
Tuesday - Camphier Town - Shock Ribbon
Wednesday - Geosenge Town - Downcast Ribbon
Cafes - Going to different cafes each day, you can meet people that will show you different pokemon to fill up your pokedex.
Lumiose City - In Lumiose City there are 3 different events, you can participate in the Lotto ID. This is the daily lottery drawing like in prior games. The more Pokemon you have traded with different OT numbers, the better chances you have to win better prizes. Go to Richissme Hotel and participate in different jobs to earn money. And go to Cafe Action to get a new statement for your PR video.
Camphrier City - There is a house northwest to the fountain, a man is inside who asks to see a certain type of Pokemon. Show him that type and he will give you a choice of berries: Cheri, Chesto, Pecha, Rawst, and Aspear.
Ambrette Town - Outside of the fossil lab, there is a "punk" looking guy who will trade any one of your PokeBalls for a Dive Ball.
Cyllage City - Next door to the Pokemon Center, there is a man who will massage one Pokemon per day to increase happiness.
Courmarine City - There are 3 different things you can do here each day. 1 - Down by the stands where you can buy the incense, the second stand has a berry you can take. 2 - For only 4 days, there is a girl who will quiz you to get 4 different TM's, she is near the coastal side of the city. 3 - After defeating the Elite Four, Tierno will ask to see a pokemon with a certain move, show him it and he will give you a heart scale.
Geosenge Town - Trevor will be here after you defeat the Elite four. He will ask if you captured a certain pokemon and tell you if it has a mega evolution.
Laverre City - There are 2 different people that ask to see Pokemon of different sizes, they are near the east side of town. You can get Poke Dolls for showing them the correct Pokemon.
Anister Dial - The only time you can find certain Mega Stones is from 8-9pm. You have to go to Anistar's sundial to after you beat the elite four and get the absolite from your rival. You will speak to the professor to start this feature.
Anister City - Go inside the Pokemon center at different times throughout each day. There is a woman who will give you a new TM each time.
Route 18 Inverse Battle - You can fight the psychic inver guy each day in an inverse battle. After a few wins he will start giving you berries.
Kiloude City - Last but not least... fight your rival.
Best Answer
Just press home. That will take you back to the main 3DS menu, and there's a clock in the upper-right corner of the top screen. There's no in-game way to see the time, but the in-game time is identical to the time on your 3DS.