The first thing you need to consider is WHAT you will be doing when you get to Bariloche, the climate (which is very changeable, specially in spring) and how long you will stay here.
If your plans are just getting around, enjoying the sceneries and seeing the country you travel to, you can wear casual, comfortable clothing.
No matter the time of the year, the key when travelling to Patagonia is light but warm and waterproof clothing. The wind in Patagonia is always cold and, even in summer, nights are chilly.
Comfortable shoes to walk long distances are mandatory and a good hat is always welcome. Sun-block is very important as well as checking every day for UV indications. But do not worry too much, Bariloche offers a wide range of shops and rentals where you can get mountaineering equipment.