Route 206 is made of two parts: the Cycling Road and a grassy area underneath. The Cycling Road is the only part of the route accessible from Eterna City, and you are not permitted to access it without a Bicycle. Here you will also find Dawn/Lucas's father who will give you the Exp. Share when you talk to him, as long as have seen at 35 separate Pokémon; if you haven't yet, make sure to come back and get it when you have.
There are a number of Trainers to fight on the Cycling Road - defeat them all and exit through the southern gate, speaking to the woman in the gate on your way to get an accessory item. Enter the grassy area by using Cut on the tree to the right of the gate. A lone Hiker has wandered down here - defeat him, as well as gathering all the items, on your way up north to Wayward Cave.
All trainers in Wayward Cave are positioned such that you could face them in Double Battles. Additionally, there is a girl called Mira in the cave who will serve as your tag battle partner. If you wish, you can find Mira before facing the Trainers to make these battles easier, since, like Cheryl earlier, she'll heal your team completely after each battle.
To get to Mira, use Flash, and, smashing your way through any rocks that pose an obstacle, go north through two 4-way intersections, east at the T-juntion, north at the next turning, then east at another T-junction, and follow the passage continuing south through a 4-way intersection, following the passage to a final T-junction, where you should turn north to arrive at Mira in the north-eastern corner of the cave. Talk to Mira to have her accompany you.
As long as you can avoid the entrance, Mira will serve as your partner for whatever double battles you wish to participate in. If you have taught Thief (found in Eterna City right beside the Eterna Galactic Building) to one of your Pokémon you can use it on wild Bronzor, since they have a 5% chance of holding a Metal Coat which not only strengthens Steel-type moves, but is also used to evolve Scyther into Scizor and Onix into Steelix. As Mira will heal your Pokémon after every encounter, and there is a chance you could encounter two Bronzor in a battle, this is an ideal time to get this item.
Once your business here is finished, head back outside and to the west, as there is another cave to explore.
To access the hidden area of Wayward Cave, start just south of the top right pillar that supports the Cycling Road, walk 9 steps west, and then go north until you reach the entrance. You won't need Flash here, except for the very small entry area, which you can find your way through easily. Notably, this is the only place in the game to contain wild Gible. You can also find the powerful TM26 (Earthquake) here by traveling all the way through the path and going up the stairs to another hidden area on the 1F of Wayward Cave. You'll need your bike to jump the ramps; use 3rd gear (the slower gear) to land in the spot directly after a ramp, and 4th gear (the faster gear) to land 3 squares after a ramp.
When you've got everything you want from this area, head south to Route 207.