In White forest there is the White Treehollow where you can
actually battle a lot of trainers until you beat them all.The White
Treehollow is also known as Unova Challenge.It is a pretty
challenging place.It has 10 floors.Each Floor with its own
difficulty.Every level has a boss that you must beat to continue to
the other floors.The 10th Floor boss is Alders Grandson Benga.The player cannot enter the Treehollow if he/her has not defeated Alder in Flocessy town.When
you beat him, go to Alders place in Flocessy town and talk to Benga.He
will give you a Shiny Dratini as a Reward for beating the Unova
Also in White Forest there Shops that sell Evolution stones.They open
according to the Treehollow Floor you are.There are people with their
Pokemon that you can talk to.You can heal your Pokemon at the Pokemon Center.
This Guy you say will leave when you beat Alder in Flocessy town.
These are pretty much all the things you can do in White forest