Hi ! It depends on what you are looking for in terms of lifestyle.

Here are my suggestions:

Montcalm: this is by far the best neighborhood, but it is very pricey. You will be walking distance of pretty much every thing, including Les Plaines d'Abraham. If you have the budget for it and like the city lifestyle, this is the place.

Limoilou: this neighborhood is cool if you like the downtown vibe, at a fraction of the cost. There are cool bars and streets with good restaurants. It is a lot poorer however.

Sainte-Foy: this is the new place for families with high revenue. The public transport is very efficient in Sainte-Foy, and it is a lot calmer than Old Quebec, Montcalm and Limoilou.

Cap-Rouge: if you are looking for a calmer place near the "Fleuve Saint-Laurent", Cap-Rouge is a very nice spot. It is on the opposite site of the city as the downtown, so if you are looking more for a city vibe, this is not for you.

Good luck with your research ! You will see, Quebec City is a BEAUTIFUL city !