Chargestone Cave is to the west. As you are about to exit the city, Bianca comes running along and challenges you to a battle.
Recommended party: Throw/Sawk/Timburr, your starter, the elemental monkey that you received from the person in the Dreamyard, and Liepard.
Bianca has four pokémon with her: Herdier, the starter that is weak to yours, the elemental monkey that is strong against your starter, and Musharna.
Bianca's L26 Herdier knows Helping Hand, Take Down, Work Up, and Crunch. Being a Normal type pokémon, Fighting moves are super-effective against it, and Fighting-type pokémon are strong against it. Rock and Dark pokémon are also strong against one of Herdier's moves, so if you have one that knows a Fighting move, that would also be a good choice.
Bianca's L28 starter (Dewott, Servine, or Pignite) should be easy to defeat with your own starter if it is at or above level 28.
Bianca's L26 elemental monkey (Pansage, Pansear, or Panpour) should be easy to defeat with your own elemental monkey that you received from the person in the Dreamyard as long as it is L26 or more.
Bianca's L26 Musharna only has a Psychic attack, and all of its other moves are status moves. It can inflict sleep on your pokémon, and it can use Lucky Chant, which prevents critical hits for a while, and it knows Defense Curl. Since Dark pokémon are completely immune to Psychic attacks, bring along a Liepard. Musharna won't be able to hurt it, but the battle could drag on for a while if you let Musharna use its status moves. Try having another pokémon paralyze or poison it before switching to Liepard.
After defeating Bianca, you will get the HM for Fly! Now you can finally return easily to the towns that you have visited already. You might want to stop by Nuvema Town to talk to Professor Juniper, because she will give you items depending on how many pokémon you have recorded in your Pokédex. And speaking of your Pokédex, you might want to look through it now for pokémon that you haven't caught, and check their area data to find out if there are any that you missed and can go back to catch.
Now that you have defeated Bianca, you can head on to Route 6. You could have checked it out before challenging the Driftveil Gym, but here is how to get through it if you haven't visited it yet.
When you first get onto route 6, look for a path in the trees above you. Walk through this path to find an item.
Continuing on the way, you find a bridge over some water, then you encounter a grassy area. Here, there is a scientist with an L27 Deerling. It is a Normal/Grass, so it is weak to Fire, Ice, Fighting, Poison, Flying, and Bug moves. Also, it is unique in that its appearance changes depending on what season it is. You can catch wild Deerlings in the grass around here.
There is another bridge near here, and to the north is a Pokémon Ranger with an L27 Emolga. This Emolga can be tough because it will use Double Team a lot, which makes it hard for you to land a hit. If you have an Excadrill, use that, but if not, then go for Sandile. Sandile might be the best choice if it knows Torment, because you can slow down Emolga's use of Double Team.
On the bridge nearby is a Parasol Lady with four L24 Tympoles. Beyond her is a Scientist with an L27 Deerling.
Now you'll find a house that has a Hyper Potion in the yard. The Scientists in the house are researching the seasons. One scientist will ask you to show him a Deerling. If you do, he will go catch his own Deerling of the same type. You can exit and go back into the house to see the scientist's Deerling walking around. Show him a Deerling every season so that he can see all of the Deerlings from all of the seasons.
South of the house you can get an Elixir from the grass.
To the west is a Scientist with an L27 Deerling. West of that there are some items in the grass, but some are actually wild Foongus, so be careful. The Parasol Lady here has an L26 Vanillite and an L26 Frillish. Then there is a Pokémon Ranger with an L27 Minccino and L27 Emolga.
A hiker will be on the bridge, but he's not a trainer. Go into the house and talk to the woman to heal up your pokémon. The little girl will give you a Shiny Stone.
Outside, when you go north to the cave, Clay will arrive and have his Krokorok cut down the yellow Galvantula web. He will also give you the TM for Bulldoze that he promised you. Now you can go into Chargestone Cave.