To many people this will sound like an oxymoron as the “best” neighborhoods are generally all in Taipei. Why do you want to limit yourself to Xinbei? If it’s because of work or your wife’s relatives, then, above all, it’s probably a good idea to look for a place not far away in terms of distance. Commuting in Xinbei can be time-consuming, especially between different districts (rather than to and from Taipei).
That being said, the “best” district as a whole is probably Xindian, especially the Bitan area. In Banqiao, your first choice could be the planned area around the train station, which is generally nice, but note that it ends quite abruptly and everything away from it is not that nice anymore. Zhonghe/Yonghe may be convenient (if you work there) or budget-friendly, but I’ve never heard it described as nice. And sorry but Tucheng is sort of a dump even by Xinbei standards.
There could also be some nicer areas farther afield, like the new developments in Sanxia around the Taipei University, in Linkou or in Sanchong. And some of the “districts” of Xinbei “city” are actually remote, rural areas (Sanzhi, Wulai, etc.), which might be very liveable as far as the environment is concerned (clean air, etc.), as long as you don’t need to commute regularly (e.g. work from home), and you have a car for errands.
Generally though, I wouldn’t expect Australian-style suburbs here; it’s all a never-ending city, just with much poorer level of infrastructure than in Taipei proper. People live in Xinbei because it’s much cheaper to buy an apartment there but since you’ll be renting anyway (I guess), the difference in rent between places in Taipei and the “nicer” areas of Xinbei is negligible (just a thought).
Good luck with the choice!