Professor Juniper will give you an Ultra Ball in the Route 4 passageway, followed by an explanation of why everyone was chosen for the adventure. As you enter Nimbasa City you will see some Plasma members trying to steal Pokemon from the Day-Care Man. Naturally, this leads to a battle.
The grunts leave and say they are going to hide in the amusement park. The Day-Care Man gives you the Bike, one of the most useful items in the game. Select the Bike in your bag and register it so you can hop on by pressing Y. This makes it quick and easy to get on your Bike. After the Day-Care Man leaves, Bianca shows up and says she is going to see a musical. Head over to the northeast corner of the city to see Bianca standing in front of the Music Theater. She will take you inside and you will meet the owner who will give you a Prop Case. The Prop Case holds decorations for your Pokemon to wear. You are given the chance to decorate a Pokemon of your choice. Once you're finished, talk to the girl east of the front desk to get TM 49 Echoed Voice. This move gets more powerful each turn you consecutively use it.
You can participate in the Pokemon Musical if you want, but it isn't that useful or exciting. It is like the spiritual successor to the Pokemon Contest in Ruby, Sapphire, and Emerald. You can dress up your Pokemon and press the "Appeal" button at a strategic moment to get the crowd on your side. There aren't any items that you can get other than more accessories for your Pokemon to wear. Because of this, I'm not going to delve too deep into the Pokemon Musical.
Walk outside when you're ready. You will enter a conversation with Bianca and her Dad. He insists on bringing her back home to Nuvema Town, but Bianca wants to continue with her adventure. Elesa approaches and persuades Bianca's Dad to reconsider. It plays into the classic adage, "If you hold a butterfly too tight, you might crush it." It's nice to see the Pokemon games give an attempt at emotional ties to the characters.
West of the Music Theater is the passageway to Route 16 where you can get a Macho Brace from an Infielder. East of the Music Theater is the Big Stadium. Depending on the day, you can battle a Linebacker, an Infielder, or a Striker once per day.
Head to the southwest corner of Nimbasa City if you need to heal your Pokemon. The building west of the Pokemon Center can't be used until after you beat the game, so don't bother entering it for now. Enter the building northwest of the Pokemon Center and talk to the blue-haired guy on the first floor to get HM 04 Strength. Strength allows you to moves large boulders, but not until you get the Bolt Badge from Elesa. Upstairs in this building is an old lady that will give you a Soothe Bell if the Pokemon in the first slot of your party has high happiness. A Pokemon that holds the Soothe Bell will gain happiness faster than normal.
The building directly north of the Pokemon Center has a blue-haired guy upstairs that will give you a Sun Stone. You can use that to evolve a Cottonee or Petilil if you caught on ein Pinwheel Forest.
The building east of the Pokemon Center is the Battle Subway. As you try to enter, a blue-haired girl will give you the Vs Recorder. The Vs Recorder keeps records of the Battle Subway and can save recordings of Pokemon battles. There are a lot of trains down here. When you ride one, you will battle trainers in succession in order to earn Battle Points (BP). You can use BP to buy things like TMs and other rare items. Unfortunately, not all of the trains can be ridden your first time in Nimbasa so make sure to come back here later on.
Walk east from the Battle Subway to enter the Amusement Park. Keep walking east to find N. He leads you to the ferris wheel. You both ride and he divulges some details about Team Plasma. Some grunts show up. He wants to battle you to let his grunts flee.