Follow the path in Acuity Lakefront to reach the Lake Acuity's entrance, and you'll see your rival Barry there. He tells you that to climb the slope that leads to Lake Acuity, you will need to win the Badge at the Snowpoint Gym. After he leaves, head east to Snowpoint City.
Snowpoint City has a few attractions, the main ones of interest to you on your first visit will be the Gym and possibly the Move Tutor.
Located in the upper right hand corner house is the Move Tutor where much like the one on Route 212 you can trade him shards in exchange for learning new moves if that Pokémon you want to teach it to is compatible.
In the upper left hand house is a girl who will trade a Haunter named Gaspar that will hold an Everstone (which prevents it from evolving into Gengar) in exchange for a Medicham.
The secret to the puzzle of the Gym is to smash the snowballs on the ground is to slide in from above at a fast enough speed to smash them. Your main goal is that from the entrance, head straight to reach the Gym Leader, but only after you've smashed the snowballs that were in the way. Just slide around the Gym and predict which areas you need to slide down to smash those snowballs. Of course it will take some time to experiment until all the necessary snowballs are gone. You do not have to face all of the Gym Trainers in this Gym.
Once you've beaten Candice you will receive the Icicle Badge that will will allow you to use Rock Climb outside of battle as well as TM72 (Avalanche) that while it causes your Pokémon to usually move last, it's power will double if that Pokémon was struck by a damaging move first.
Now that you can use Rock Climb (optional)
Now that have access to the move Rock Climb in the field, we can gain access to some items and fight some trainers around the region.
When you arrive at Lake Acuity, Galactic CommanderJupiter has already defeated Barry and caught Uxie. She leaves while stating that Barry is a weak Trainer. Barry, though upset, states his desire to become stronger. Surf across the lake to the patch of grass in the northeastern corner, where you can pick up TM14 (Blizzard).
Head towards the Team Galactic HQ in the northeast corner of Veilstone City and talk to the lone grunt there, who is angry at you since his Clefairy was taken away after you defeated him before. He claims ignorance about a Storage Key and runs away. Luckily, Looker shows up and states he has the Storage Key with him. He suggests that it is best to enter the HQ through the Galactic Warehouse you were at before. Make sure to select a well-balanced team for this mission, as you are going to need it.
Trainers with a Vs. Seeker by their names, when alerted for a rematch using the item, may use higher-level Pokémon.
Head to the metal doors you saw before and Looker will arrive and open them for you. Once that is done, continue down the stairs to the basement of the warehouse. This bottom floor connects to the Team Galactic HQ. Follow the hallway east to the stairs at the end, battling the two grunts.
Once up the stairs, you'll arrive in the very upper right-hand corner of 1F. After defeating the Galactic Grunt there, use the right warp panel to enter a closed-off area where you will find TM49 (Snatch). Step back onto the panel from whence you came. Afterwards, use the left warp panel to go up to 2F.
Galactic HQ 2F
From here, use the nearby warp panel to return to 1F.
Ignore the grunt to the right and take the panel to the left, go down the stairs and defeat the Scientist, grab the Dubious Disc, take the second set of stairs to the left and keep going west.
Grab the Galactic Key, which will allow you to open the locked steel doors, and you will meet a Galactic Grunt who mentions that Cyrus is having a meeting. You can get to it by entering the HQ's main building from Veilstone and using the Galactic Key. You can unlock the nearby door which will allow you to head back easily to the Warehouse's entrance, but let's make a little side trip by backtracking to 2F. Use the stairs to access 3F.
Team Galactic HQ
Galactic HQ 3F
Defeat the Galactic Grunts and then open the locked door to find TM21 (Frustration). Now head back to the Galactic Warehouse and leave there to return to Veilstone City. It is now time to enter the main building.
Team Galactic HQ (main building)
Use your new Galactic Key to open the locked door and head up to 2F.
You should now be in the middle area of 2F. Simply fight the grunts and then use each of the warp panels. This will earn you two items. Afterwards, use the warp panel that is located by the TVs to reach the meeting hall, where Cyrus is.
Galactic Meeting Hall
You will meet up with Looker here and listen to Cyrus giving his speech about how the world is incomplete and the fact that he now has the power to create a new one for Team Galactic's benefit. After he and the grunts leave, Looker leaves to investigate this further. Follow in the direction he went to find another warp panel that will take you to a new area of 2F.
You'll be in the upper-right area of 2F where there are some beds to rest your Pokémon team. Hidden in the right-hand bed is a Poké Doll. Climb the stairs to 3F.
After dealing with two Galactic Grunts you'll find three warp panels here. The upper right one takes you to an area where you can find a Protein. The upper left will get you a Max Elixir, and the lower left is the one you'll need to proceed to an area with stairs up to 4F.
Galactic HQ 4F
This is where Cyrus's office is located. After you open the locked door, Cyrus will greet you and mention that the new world is actually only for him, since he feels his underlings are just as flawed as the rest of the world. Finding you interesting, he'll challenge you to a battle.
His team is the same as the previous, except that two of his Pokémon have evolved. His first Pokémon, Sneasel, is easily vulnerable to Fighting-types. His other two Crobat and Honchkrow both are Flying-type and should be easily vulnerable to Rock, Ice and Electric-types. Just remember that Crobat is the faster of the two and may try to confuse your Pokémon with Supersonic, and Honchkrow possesses better physical attack moves that could KO your Pokémon if you are not careful.
Lake Trio room
Afterwards, Cyrus will give you a Master Ball and then leave. He mentions that if you want to save the Lake Pokémon, you should use the nearby warp panel to get to them. Use that warp panel and you will reach a hallway. At the end is where the Lake Trio are being held. Once you reach that room, where they are, you can overhear a conversation between Charon and Saturn about how Charon is happy with the new world Cyrus will create. Talk to Saturn and he'll challenge you to a battle since, unlike Cyrus, he will not let you roam freely and interfere with their plans.
Saturn's team is the same as before but at higher levels, so the same tactics you used at Lake Valor should work here. Once you defeat him, Saturn will let you press the switch that will set Azelf, Mesprit, and Uxie free. He mentions that Cyrus crafted a special Red Chain for capturing something on Mt. Coronet.
Return to Cyrus's office and a new pathway will have opened up with a warp panel that will take you to the green warp panel just inside one of the main doors of the Galactic HQ. Take it, exit the building, and heal up at the Pokémon Center.