Fog Badge - Pokemon Soul Silver
Head North to Route 35. You can continue North to the National Park but the game continues to the East. Cut down the tree and to reach Route 36.
You will come to a tree blocking your way. You need to use the Squirt Bottle on it and it will transform into a lvl 20 Sudowoodo. This is your one and ONLY chance to get him so if you are a collector work it down and then Poke Ball for the win. After beating it a girl will come and give you a Berry Pot to grow up to 4 Berries at a time. After that is West and North to Route 37.
This is just a short Route filled with Trainers on your way to Ecruteak City. You can get some good Fire Pokemon then on to town.
Head into the house that is to the left of the Center. The guy in there will give you the Dowsing MCNH. It assists you in finding hidden items. Then into the Theater above the center, fight the guy on stage and afterwards the man in the audience will give you Surf. You can't use it yet, but you'll need it soon enough. You can't hit the gym yet so head to the Burnt Tower.
Eusine will greet you as you enter the Tower and talk about his quest for Legendary Pokemon.
lvl 20 Gastly, lvl 22 Croconaw, lvl 20 Zubat, and lvl 18 Magnemite vs. lvl 28 Quilava and lvl 20 Flaaffy
As you head to the ladder your Rival will challenge you yet again. This is not as hard a battle as some of the previous. Just make sure to switch out your main with something that is powerful against his main.
When you head downstairs the 3 legendary Pokemon with run away and Eusine will come down and talk about how awesome they are some more.
Fog Badge - Burnt Tower
You have to fight all the trainers. As soon as you beat them their light goes out making it hard to navigate. If you fall off though there is no penalty and their lights go back on. So no real problems getting to Morty.
lvl 21 Gastly, lvl 25 Gengar, lvl 21 Haunter, and lvl 23 Haunter vs. lvl 31 Quilava, lvl 19 Dratini, lvl 21 Flaaffy and lvl 21 Sudowoodo
Be prepared to be put to sleep A LOT. All of Morty's Pokemon like to throw Hypnosis your way so be prepared. Afterwards you'll get the Fog Badge and you can now use Surf to go across water.
Fog Badge - Badge Battle