Follow the rather straightforward path, walking through grass in some areas and battling against trainers. You don't have any pokéballs with you now, and you have no way to get any yet, so you can't catch the wild pokémon that you see.
Soon, you will reach Cherrygrove City. The Guide Gent will show you around the town, and afterward, he will apologize for running around so quickly. To make up for it, he gives you the Running Shoes. These are automatically a part of the touch screen from now on. You can hold the B button to run around, but you can also tap the orange Running Shoes tab on the touch screen to turn the running shoes on. This way, you will run without needing to hold down any buttons. You can turn them back off by tapping the orange shoes tab again.
Heal up your pokémon in the Pokémon Center. You will be doing this a lot throughout your adventure! You can buy a few things at the pokémart, but not any pokéballs at this point. I guess this is to keep you from getting distracted from the task at hand: finding Mr. Pokémon.
When you head north to continue on your journey, the Guide Gent will stop you for a second to give you a map card for your pokégear. This is a map of Johto that will show you all of the routes and major locations. It has many useful features, including allowing you to make notes for each "square" of the map. When you're ready, go north to Route 30.
Along this path, the guy in the house will give you an Apricorn Box. This will store apricorns, which you can get from apricorn trees. There is one next to the guy's house: a green apricorn! Check on the tree to shake it, making the apricorn pop out. A new apricorn will grow every night at midnight, so you can keep checking on these sorts of trees every day to increase your apricorn supply. They have good uses, but you can't use them just yet, so for now you should just collect apricorns when you can.
As you go along, you will find another house, this time with a pink apricorn tree next to it. Get the apricorn and then go inside. It's Mr. Pokémon! Professor Oak is there, too, and he gives you a Mystery Egg. You will be healed up, and then he will give you the highly useful Pokédex. This is like an encyclopedia of pokémon. Whenever you see a pokémon, an entry automatically gets added to the pokédex, along with the locations in Johto where that type of pokémon can be found. When you capture a pokémon, more information for that type of pokémon gets added, such as the pokémon's type, size, and so on.
The pokédex gives you a good way of finding and catching a pokémon that you have only seen when fighting a trainer. You can look at the pokédex entry at the areas where that pokémon hangs out, and then go there and catch one yourself!
Oak will give you his phone number, and then you can go outside. Professor Elm will say that something bad happened, and asks you to come to his lab. So do that!