Fly to Sandgem Town and use the south exit to reach Route 219. It is the first of a chain of routes that you can use to reach Sinnoh's Pal Park. The Pal Park will be inaccessible until you have the National Pokédex, but the training you'll get on the way there is valuable, and the Pal Park can be flown to after you visit it once. You'll want to bring Grass- and/or Electric-types for the trip since the Trainers on the upcoming Routes use mostly Water-types.
Though not as simple as Route 219, Route 220 is pretty straightforward. Head down the eastern side when the route forks in two, fight the Swimmer there, then backtrack and head down the western fork. A little further on, the way forward forks in two again, this time around a largish island. Grab the Max Repel just north of it, then head down the southern fork until you reach the island with the submerged shoreline. Continue east past the Carbos, making a quick detour into route 221 to fight the female Swimmer, then backtrack and head north where the Carbos is, head west briefly to fight the male Swimmer, then go east to Route 221.
Continue east from Route 220. Fight the male Swimmer, and continue east until you reach land. Head east on the northern section Route 221, stopping at the Level House (see below) to pick up the Pure Incense which allows you to produce a Chingling Egg when breeding Chimecho. When you reach the Pal Park (inaccessible until after you defeat the Elite Four), head back along the southern part of the route, collecting all the items you missed when travelling on the northern part. Now that you're finished here, it's time to Fly back to Jubilife City and head northwest via Route 218 to reach Canalave City.
The only house located on Route 221 belongs to an old man who will each ask for a Pokémon at a certain random level from 1 to 100 each day. The items he'll give you are the Black Belt, Expert Belt, or Focus Sash. Only one item is given to you out of that particular order when you complete his request, and after he gives you the Focus Sash, the cycle will start over.
For days where the requirement is an extremely high-leveled Pokémon, you can try soft resetting and not much else. On some days you might get lucky and be able to use Pokémon from your regular team. For days where low-level Pokémon are needed, the quickest solution is to simply head to an earlier area with an Quick Ball or two, catch a Pokémon of the appropriate level, and bring it to him.
Now that you have Surf, you're able to navigate the rest of Route 218 and reach Canalave City, your next destination. There isn't much to do on this short route except for fighting the Trainers and catch Mr. Mime (this is the only location in Sinnoh where they can be found).
When you enter the guard post separating Route 218 from Canalave City, Professor Rowan's assistant will approach and give you upgrade to your Pokédex. This will allow you to check the differences in appearance between the same Pokémon in their Pokédex entry, such as gender or form differences.
When you first enter Canalave City, be sure to visit the Pokémon Center to heal and prepare your team - when you try to cross the bridge your rival Barry will show up and challenge you to test your skills.
Barry now has a Heracross on his team, but a powerful Flying-type move can easily beat it. Unless you have an especially fast one, avoid using Psychic-type-types as Heracross's Dark-type move Night Slash will counter it. Four members of Barry's team have reached their highest evolutionary form, but they shouldn't be too much of a problem as long as your party is at least at Level 35 and has good type coverage. After the battle, Barry will suggest that the player visit Iron Island before heading off towards the library.
The Move Deleter resides in the house just north of the Poké Mart. He will be able to remove any unwanted moves in your Pokémon's present moveset including HM moves. This is the only way for a Pokémon to forget a HM.
The southernmost house on the west side of town belongs to a Sailor named Eldritch, who handles the ferry in town. Nothing important will happen here until the post-game after obtaining the National Pokédex. Eldritch is waiting near his boat at the dock nearby, and he can take you to Iron Island which will be required to visit. South from the wharf is a hidden area where you can pick up the TM89 (U-turn).