Everyone is going to say Luzern. Everyone. OK, almost everyone. If you do a search of the forums you will see the overwhelming feelings. I agree with them all. Skip Zurich if your time is at all limited.
That being said, a lot of people also argue that Zurich is worth seeing. I agree, there are pockets of Zurich that are worth visiting. Listen to the people who have been there. There is a bar I like, a bee store right outside of town, a water park that is an education, and Bremgarten next door (and towards Luzern). Other than that I just don't get it.
Stop by Bremgarten for lunch or dinner. Eat something light at the Bijou (http://www.bijou-bremgarten.ch/) and head around the corner to Mamma's or the Bodega for something to eat. Really decent food there. The Bijou is beautiful as is its view. Superb, especially for city life ;) I came to it via a footpath with a local my first time and didn't know until my second visit how awfully big it is. I thought it was was just some seats & tables under an awning with a great view of the river--HA!
As for Engelberg, see the cheese 'factory' and, if you are sick, the doctor in the same building. But for something REALLY Swiss, take the lift up, above Engelberg, to BrunniHuette (link to the area, http://www.brunni.ch/en/food-accommodation/brunnihuette-sac/). You can go up for a little while (lunch, or the afternoon, or a hike), the day, or for overnight. If you are staying longer, you can go to the right, on foot, for about 4 hours, to Rugghubelhütte. Take your boots with you for that one, but BrunniHuette is right at the lift, as is the pond behind it. It is sweet and ripe with good views, tasty chow, and cold beer and wine (and ice cream for the tots).
I have never even been through Lugano I don't think...
