A romantic weekend with your girlfriend? Well, what can be more romantic than taking a long walk in Parco del Valentino? Here's a picture from my last visit to Orto Botanico, just to give you an idea of what I'm talking about:
Just walking around the Centro (the city's centre) is a fulfilling tourist experience, the city's architecture is its main attraction. I won't bore you with details that you can find in any tourist guide, the only thing that I must point out is that if you decide to visit Turin you shouldn't miss the Museo Nazionale del Cinema. It's a little bit more than its name reveals and far from limited to Italian cinema. I've spend about 6 hours exploring the various exhibits, and revisited the next day (I was foolish enough to ignore the panoramic elevator the first time).
Since you live in Lyon, you might have heard the medieval legends that link the city to Turin, the two cities along with Prague are supposedly the three points of the "international triangle of white magic". One of the more interesting stories I heard from locals was about the Monumento al Traforo del Frejus, at the end of Piazza Statuto, the figure at the top of the monument is said to look at the direction of Lyon. All this may not be your cup of tea, but I must admit the various stories were a great source of light entertainment during my first visit.
Turin's nightlife is... not its main attraction. You'll find several small cafés and bistros in the Centro, but your options after about 11pm are extremely limited. If you were visiting during the summer, the bars on the Murazzi would be your better option, during winter... I have no idea ;)
I haven't visited Genoa, but given the choice, I'd probably re-visit Turin, just for the architecture alone.