Where is Burgh's Gym?
Burgh's Gym (Castelia Gym) is located in the westernmost street of Castelia City. The entrance is on the right side, with the pokéball statues on either side of the door, and a pokémon Gym symbol above the door.
Since you will encounter Bug type pokémon in the gym, you should try to catch Fire-type pokémon. If you go north from Castelia City into Route 4, you can find Darumaka in deep sand. Also, back in Pinwheel Forest, if you walk around near grass, you might get a shaking spot, which has a chance of being a Pansear. If your starter is Tepig, bring it, too.
Another good type to bring is Normal/Flying, like Pidove. Fighting types are also strong against Bug moves, so Throh, Sawk, and Timburr can be part of your party.
If you still have open slots in your party, try Roggenrola (from Wellspring Cave).
Getting through the Gym Puzzle
Burgh's gym is somewhat like a honeycomb. There are honey walls that you can walk through by walking straight up through the middle (the sides just bounce you back, so go through the center) until you emerge on the other side. There are also switches on the floor. You need to press all of these to proceed, because they control the metal gates that block some of the honey walls. However, some switches reveal a trainer instead of opening the path. Once you have opened all of the gates, you reach a purple area with stairs leading up to Burgh.
Getting through his Gym is pretty straightforward: go through a honey wall, press whichever switches you can reach, and repeat, until you get to Burgh.
After you unlock the first gate and go through the honey wall that was behind it, the switch just beyond this honey wall will make a trainer appear instead of opening a gate. This will happen again if you press the switch at the far left of the gym.
Strategy for the Trainers
Each of the trainers here has a Sewaddle. Three of them are L20 and one is L21. These are super-vulnerable to Fire and Flying moves, so use Darumaka, Pansear, Tepig, or Pidove.
You will also encounter a couple of L20 Venipedes. A good opponent here is a Timburr that knows Rock Throw, or a Roggenrola. A Fire type is not suitable here, nor is a Flying type, because they are vulnerable to Venipede's Rock move.
One trainer also has an L20 Dwebble. Being a Bug/Rock pokémon, it is weak to Water, Rock, and Steel moves. Its moveset will possibly include Bug, Dark, and Rock physical attacks. As such, Fighting type pokémon have the best defense against it. A Timburr that knows Rock Throw will work well here.
Strategy for Burgh
Burgh has three pokémon with him. All three know the Bug-type special attack Struggle Bug. This attack reduces the Special Attack of the hit pokémon and also does damage.
Burgh's L21 Whirlipede knows Struggle Bug, Poison Tail, Screech, and Pursuit. Because of Pursuit, you will want to make sure that the pokémon who is first in your party is the one that you intend to have fight against Whirlipede, so that you won't have to switch it out and risk it getting attacked. As with Venipede, a good opponent is a Timburr that knows Rock Throw, or a Roggenrola.
Burgh's L21 Dwebble is up next. Dwebble knows Struggle Bug, Smack Down, Sand-Attack, and Faint Attack. A Timburr that knows Rock Throw is the best opponent here.
Burgh's L23 Leavanny is the last one. Leavanny's attacks are Razor Leaf and Struggle Bug, and it knows Protect and String Shot. A Fire or Flying pokémon would work well here. These types are strong against Leavanny's attacks, and they also have the types of attacks that Leavanny is extremely weak to: Fire and Flying. The pure Fire-types mentioned above would do well, as would Pidove.
When you win, Burgh gives you the Insect Badge and the TM for Struggle Bug.