Some suggested activities:
Kite board (Shippagan Lameque Island)
Golf in Pokemouche
Beach
Kayak

Festivals you do not want to miss:
FestiVin Wine Festival (June)
Caraquet Lobster Moto Weekend (July)
Folk Tales of the World Festival (Historical Acadian Village)
Visual Arts Festival of Atlantic Canada (July)
Gala de la chanson (Singer-Songwriter competition in July)
Acadian Festival (August 1st to 15th)
Show and Shine Antique Car Show (September)
Agricultural Fair, Historical Acadian Village (September)
Blues Festival in Caraquet (September)
Acadian Poetry Festival (October)

For more information about activities
in the Acadian Peninsula: http://ville.caraquet.nb.ca/

You can also consult the official tourist guide of New-Brunswick