Now that you have the Cobble Badge, make your way north of town towards the Galactic warehouse. You'll find Prof. Rowan's assistant at the entrance. His/her Pokedex has been stolen by Team Galactic. You'll team up with him/her to battle the Grunts in a double battle. They have four Pokemon, all level 25, so you shouldn't have a problem wiping them out. After you win, the Grunts will run into the warehouse. Enter the warehouse to obtain HM02 (Fly) and exit. Your next destination is Pastoria City, but the fastest way there is by Hearthome City, so head there first, then head south to Route 212.
Route 212 is fairly linear filled with quite a few trainers along the way. The first trainer you would encounter is a Policeman, but he only challenges you between 8pm and 4am. The Pokemon Mansion along the way is a good place to catch a Pichu and Pikachu, and it's a hotspot for certain rare Pokemon after you beat the Elite Four. After you're done checking out the Pokemon Mansion (or if you don't bother with it), keep heading south, then east to reach Pastoria City. Be sure to have your bike handy along the route too, because it's the only way to cross the logs along the route.
Firstly, go to the Pokemon Center to heal. This town is the location of the fourth badge you need. You can visit the Great Marsh, which is the Safari Zone of Pearl and Diamond. Pay the fee and talk to the lady near the entrance. She will give you HM05 (Defog) which you'll need later on. After that, just leave the Great Marsh, or catch some Pokemon if you wish, since you've already paid the fee to enter.
Time to enter the Pastoria City Gym, which you will notice, is another lovely puzzle. This one involves raising the water level to advance. Start by going through the middle, then left, then down and step on the button. The rest is pretty linear, so I'll leave that challenge to you. After you make it through the puzzle with (hopefully) no troubles, you will come up against Wake, the Gym Leader of Pastoria City.
Wake has Pokemon with the same exact levels as the previous Gym leader, so if you've been doing some training between the two, this battle shouldn't be too bad, especially if you have an electric and/or grass Pokemon, then it will just be a walk in the park.
After you've shown Wake who's boss, you will be given the Fen Badge, which will allow you to use HM05 (Defog) outside of battle. You will also be given TM55 (Brine), which doubles in power if the target's HP is below 50%.