In the original Pokémon Diamond & Pearl, you could head to Amity Square and hang out with your Pokémon.
The Square makes a return in Brilliant Diamond and Shining Pearl, with a few changes -- including being able to take your Pokémon with you as you travel Sinnoh.
How to get to Amity Square in Brilliant Diamond & Shining Pearl
To access the Amity Square, you must have at least two Gym Badges, have obtained the Bike, and have unlocked the Grand Underground.
Then, head to Hearthome City. The Square is in the north area, behind the Gym. There are entrances on either side.
There are several items you can pick up here.
In the south, along the treeline, you can find TM43 Volt Switch. North of the fountain is the Poffin Cooking Station, and TM45 Attract next to it.
Head east and up the stairs, then north to another set of stairs. A visitor with a Drifloon will explain that the strange buildings are ruins from long ago.
In the ruin to the left you can find an Amulet Coin, which will double the Pokédollars received in battle if held by a participating Pokémon. To the right of the other ruin you'll find the Spooky Plate held item.
If you try to leave, the Receptionist will tell you that your Pokémon wants to go for a stroll. After this exchange, the follow mechanic will be unlocked.
To choose a Pokémon to walk with, bring up the menu with X, select Pokémon, then press A and choose Walk Together. Unlike the Amity Square itself, this works with any Pokemon.
When out walking together, your Pokémon will direct you to items. Talk to your Pokémon often to check for items, by facing it and pressing the A Button.