Mactaquac is a provincial park located about 15 minutes outside Fredericton on Route 105 (the old Trans Canada Hwy). It is a nice clean beach on the Mactaquac Head Pond. The water is usually fairly warm and very clean.
They have campsites there for around $20 per night. There is also fishing and walking trails. It's a great spot because it's very quiet, but still only 15 minutes from town, and all the fun stuff that comes with it.