Cherrygrove is the first city that you'll encounter in the region that has a Pokémon Center and a Poké Mart. Pokémon Centers are essentially hospitals where your creatures can be restored to perfect health. The PC at the rear wall is always available for Trainers to organize their Pokémon and items. Poké Marts offer a wide range of items for sale, from Potions to Poké Balls.
Talk to Guide Gent at the east entrance to get a tour of the city. After a quick jog through the city, he gives you the Map Card to add to your Pokégear. This acts as the Town Map from Generation I. To use it, press Start, select Pokégear, and move the cursor to the map icon.
In the southern house is a man who is knowledgeable about the healing properties of Berries and Berry trees, so stop in to get a free Berry. Check the tree outside for another, and continue north. A Pokémon battle blocks the main road, preventing anyone from reaching Violet City for now. Trek through the tall grass on the eastern path to reach Mr. Pokémon's house. Check his tree for a PSNCureBerry, and head inside.
When you walk inside you'll see that he has another guest, who happens to be Professor Oak from the Kanto region. Mr. Pokémon gives you the Mystery Egg to take to Professor Elm, and Oak gives you a Pokédex. The professor rushes off, and Mr. Pokémon heals your Pokémon before you leave.
Professor Elm calls when you step outside, obviously in a state of panic. Hurry back to New Bark!
On your return trip through Cherrygrove, that mysterious guy that was spying on Elm's lab ambushes you, forcing you into a battle. It turns out that he swiped one of Elm's Pokémon from the lab. Whichever you chose, his Pokémon will always have a type advantage against yours, though at such a low level it is not much of a threat.
When you return to New Bark, the police are questioning Professor Elm in his lab. After speaking with the officer—and naming your rival as the culprit—talk to Elm to deliver the Mystery Egg. He is shocked to see the discovery, and promises to call you if he learns anything about the Egg. On your way out the door, his assistant hands you five Poké Balls to use on your journey. Now that you've got your first supply, you are free to buy them in most Poké Marts.
Stop back home before you leave town again. Talk to your Mom, and she'll offer to save your money for you. When all of your Pokémon faint during battle, you lose half of your money, but any money that Mom has is kept safe. She'll sometimes buy a random item for you from your savings; it may be medicine like a Potion, or a stuffed doll to decorate your room. You can change whether she saves your money or not by phone as well.
With Elm's errand complete, the Pokémon battle on Route 30 will have concluded, so you can finally continue on to Violet City. Go north from Cherrygrove and follow the western path.