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Hey there!
Turin is a really nice city for biking. It has quite a lot of cycling lanes (not Netherland's level, but not bad) and a lot of places where to find bikes and bike repair shops.
As others said, look for Decathlon for brand new bikes or subito.it for used ones. But beware, there are approx. 1 million stolen bikes a year in Turin! The best investment you could do is a good set of lock, do not cheap out on those! I see constantly people buying 5€ locks for 100€ bikes, and I cannot but laugh about it. And don't buy stolen bikes: those have a really short lifespan, trust me.
Always lock the chassis AND both wheels AND the saddle if you want to stay sure, and if you don't live on higher floors I'd definitely recommend you to take the bike in your appartment, or at least in an inner yard. It's a lot of effort but nothing compared to buying a spare piece or the whole bike again.
I've read an interview to a bike thief, and the best recommendation is to buy an U lock, and have no chain resting on the ground (they're easier to cut).
Having said that, you could always rent a bike. In Turin, there's a fantastic service called ToBike, which costs 20€ a year and allows you to rent bikes for half an hour anytime you want, taking them from the stations situated all around the city. There's even an Android app which is super useful (also an iPhone one, if you're into that). It's really well-served in the city center, a little less on the periphery but overall it is an excellent service I really love.
If you need other infos, ask away!