The best thing to do in Quebec City is to wander around in the Old City! You'll be staying right in the heart, so that should be easy. There are little streets with outdoor merchants, lots of artist & artisans, fine jewelry shops. The Civilization Museum (right in the Old Town) usually has some interesting stuff (I go everytime). You should definitely walk to the Citadelle for a guided tour, lots of stuff to learn & see about the French-English battle, and the walled City.
As for the Montmorency Falls, they are indeed very nice. But just so you know, you'll have to leave the old town to go there and to Ste-Anne de Beaupré, and driving the short distance in/out is quite terrible & congested.
About the Shrine of Sainte-Anne-de-Beaupré, I must warn you that it has become a very commercial place, very sad indeed for a place of worship. They have installed TVs for everyone to see, etc. It is still a very stunning basilica. Whatever you decide, just do not drive there on Saturday, everyone is leaving Quebec City for the week-ends, and roads get congested.
For restaurants you have so many choices directly around your hotel, it is mind boggling. Some personal favourites: Gambrinus, right across the street from Chateau Frontenac (15$ for lunch on weekdays, 35-40$ for a complete meal at night) & l'Entrecote St-Jean, a good steakhouse, at 1080 St-Jean street. There is also the Laurie Raphael, which is very fine cuisine indeed and can run up to 100$/each for a complete 7 course meal, on par with what you'd get at Victoria & Albert (you could probably get a 3 course meal around 60$/each). This one would be by reservations only, but you should get one without 3 months advance unlike WDW! Finally, don't forget to at least try out a Poutine at Asthon, it's fries with cheese and brown sauce, a local speciality.
You could spend 2 full days exploring the Old City & visiting, and do a combination day with Montmorency Falls (great place for a picnic) & Ste-Anne de Beaupré. I don't know how much running around you want to do, or how relaxing you want your holiday to be, but there are plenty of other things in the area. Hope this helps, and I'll be happy to help more if you'd need.