Yes, Vianden is lovely and actually easy to get to. Luxembourg (country) offers incredibly cheap public transit service: 4 euros for a an unlimited day pass, on buses and trains all over the country. (Albeit, it's a small country!). I think someday it is supposed to become completely free.
You'd get a bus to Vianden, but you might need to connect via a train or bus somewhere else. I would use Google Maps now but didn't have a smart phone when I was in Luxemborug (2013) - but you can certainly figure out the buses/trains via websites online if you can't with Google Maps. I found it very easy to get around.
Luxembourg City itself wasn't my favorite city, but it's certainly worth some time to see. The park down below town in the gully/canyon is a nice walk with dramatic views (you'd get to the tunnels from there, but I did not see them). There's also some sort of market in town. But, I would most definitely get out to the countryside - Luxembourg the country is beautiful, with lush green rolling hills and pretty towns like Vianden.
Besides Vianden, you could visit Diekirch (direct train from Luxembourg City), not quite the charming town that Vianden is but still pleasant. I visited to go to the World War II museum there, which features exhibits about the Battle of the Bulge in World War II, partly fought in this area. (The museum is kind of old and retro these days but still interesting.) I also took a bus out to the town of Hamm near Luxembourg City to the US military cemetery out there. (Bus gets you most of the way, then about a 1/2 mile walk as I recall.) Most of the dead buried there perished in the Battle of the Bulge. One person buried there who did not die in this battle is General George Patton (died shortly after the end of the war), who has a prominent monument there, outsized in death as was his personality in life.