After you defeat Fantina and exit the Gym, you'll see Cynthia outside waiting. She'll thank you for the help in Celestic Town and explain that you should visit the library in Canalave City, so you'll want to do just that. The fastest way to make it to Canalave City is to go to Jubilife City, then head west and you should be on Route 218.
Before you go to Route 218, you should make sure you have teach Surf to a Pokemon on your team. You won't be able to continue unless you do. After doing such, surf through the very linear route to reach Canalave City, where your next gym leader badge is located.
Upon arriving, you should heal as always, if necessary. I would advise doing so, because you will confront your Rival when you walk to the bridge.
Your best bet with this battle is to have a flying Pokemon or a Pokemon that has any type of flying move, so you can take out his Heracross with ease. After you defeat your Rival, you will now be able to take on the Gym Leader, so head to the Gym if you are well prepared.
Upon entering, head left and up to take on the first trainer. After you defeat that trainer, take the last platform on the right. Head up and right to take on the next trainer. After you defeat that trainer, head down to the platform. When you land, go down and left to take on another trainer. After that, keep heading left then up onto the platform. When you land, go up and you should come across another trainer by a zipline. After that trainer is defeated, keep heading right and defeat that trainer, then head down the zipline and up the other zipline to the next platform. When you land, go up to take on another trainer. After that trainer, head left then down the next platform. When you land, you'll see a red platform to the left. Step on the platform to be raised all the way to the top floor and you will be in front of Gym Leader Byron.
Hopefully you have a strong fire type by this point. Use it to take out Bronzor and Steelix. When he brings out Bastiodon, take it out with a water, ground, or fighting type Pokemon.
When you defeat Byron, he will award you with the Mine Badge, which will allow you to use HM04 (Strength) outside of battle. He will also give you TM91 (Flash Cannon), which has a 10% chance of lowering your opponent's Special Defense by 1 stage.