I'm not super familiar with Sussex other than it's about halfway between Moncton and Saint John.
As for things to do in Moncton:
- Magnetic Hill & The Zoo are decent touristy things
- Pumphouse MicroBrewery & Restaurant is good
- Moncton gets some good acts through, check the Casino, Capitol Theatre, and Colliesum
- Tree-Go is pretty awesome
- Wildcasts could be playing
As for Saint John, I'm not as familiar but:
- Reversing Falls is cool (the waterfalls changes direction depending on the tide)
- Downtown is small but a nice walk to explore
- Sea Dogs could be playing, and Harbour Station gets the odd bigger show
If you're willing to drive 1.5-2 hours then Fredericton has the Beaverbrook Art Gallery, Jazz Fest *might* be going on then, but I think it may be earlier than mid-September. Fredericton is a very pretty city to explore.
Halifax is also only about 2 hours away and I think there's a thread about Halifax/Nova Scotia not too far back.
That's off the top of my head, if you have any specific questions let me know. I've spent a few years in New Brunswick, and have lots of family there.