Mystery Gift is unlocked after you receive three gym badges, or, if you have gotten the first badge (the Oreburgh Gym badge), you can return to Jubilife City and battle the Team Galactic Grunts. After that, you can enter the Jubilife TV Station, then go up to the third floor and talk to the person who asks what TV shows you like, and choose "EVERYONE HAPPY WI-FI CONNECTION".

If you are in the same map area as your secret base, it will be marked on the Grand Underground map with a big red circle. If you don't see a big red circle on your Grand Underground map, go to a different area of the Grand Underground (i.e. an area that you can't walk to from your current Grand Underground location) and check the map for the red circle. Keep trying different areas of the Grand Underground until you find the red dot, and go there to enter your secret base.

There is a Pokémon Nursery in Solaceon Town in Pokémon Brilliant Diamond and Shining Pearl where you can breed pokémon.

After you defeat the Elite Four and Champion, you can go to Snowpoint City and talk to the sailor at the dock to go to the Fight Area. After you unlock the National Pokédex, you can reach the northwest part of the Fight Area, where there is a fisher standing next to the gate to route 225. Talk to the fisher to obtain the Super Rod.