Go ahead and heal up your weary Pokemon in the nearby Pokemon Center, then try head north through town. You'll be stopped by a man who takes notice of how many Pokemon you've seen in your Pokedex. He's Prof. Juniper's dad, Cedric Junpier, and will power up your Pokedex so you can see different forms of Pokemon or something.
Right behind him is the Gym Leader herself, Skyla. She mentions something about Sinnoh, though I'm not sure what. Maybe she just got back from Sinnoh in one of her airplanes, who knows! Anyway, she points over towards the Celestial Tower to the north and says that's where Route 7 is as well. She'll leave for the Celestial Tower after some more dialogue, leaving you free to explore the city.
The Poke Mart inside of the Pokemon Center sells you some more TMs if you've got the money for them!
- TM07 (Hail) for $50000
- TM11 (Sunny Day) for $50000
- TM18 (Rain Dance) for $50000
- TM37 (Sandstorm) for $50000
None of these are must-haves for now, but Rain Dance and Sunny Day do help power up your Pokemon's Water-type and Fire-type attacks respectively (while weakening the other), while Hail and Sandstorm are also helpful. Sunny Day also causes SolarBeam to be used instantly rather than taking a turn to charge, so since you probably have the TM for SolarBeam, Sunny Day complements that move nicely. Depending on your team, you may want to get one of these, but do remember it is a high price to pay at this point in the game (although if you're stingy with money like me, you'll probably have enough to buy two without worry — I'm just saying you probably shouldn't).
Move Deleter and Relearner
The house just to the east of the Pokemon Center is a very important one for you to check out. The lady in the top-left corner of the room is the Reminder Girl, who will offer to reteach your Pokemon any moves it can learn through leveling up (up to its current level, of course). She charges you one Heart Scale for doing this, which you can obtain by using the Feeling Checker on your C-Gear with other players — or, if you're playing solo, I believe there's a girl in Driftveil City who might do this for you.
The man to the right of her is the Move Deleter, a man that will delete a move of your choice on one of your Pokemon — including HM moves, which can not be removed in any other way. He doesn't charge anything for this service.
In the "airport" (it's actually just for cargo planes I think) on your way to the northwest of town, you can talk to one of the people inside to get the Sharp Beak, an item that increases the power of the holder's Flying-type attacks by 20%.
The Gym is just past it, to the north, past the airplane. Yeah, that's right, an airplane! How awesome is that? Unfortunately, she's not here at the moment, since she went over to the Celestial Tower instead. Awwww.
At the very far south part of the runway, you'll find some storage container things. Not really all that much there from first glance, but you can pick up some items if you use your Item Finder, such as a Parlyz Heal, a Max Elixir, and a Repel. Joy.
Hug the eastern wall on the runway as you head back north and you'll find a little inlet where a mechanic is working on a plane. There's also TM58 (Sky Drop) right near him, which tosses the target into the air for a turn and does damage when it lands — sort of like a reverse-Fly.
Well, that's all of the excitement here for now, so head on northeast onto Route 7!