Luggage and bags:
Leave your luggage and bags in a safe place! Otherwise you risk that you return home with less than you brought. It's best to use a small rucksack with something to drink and eat. Avoid having valuable stuff there (see warnings and dangers!).
Clothing/Shoes/Weather Gear: No matter which seasons, I recommend the "layer technique". Besançon has continental climate which means you can experience various weather types in a short span. You may get a bad day in summer with only 10°C, but on the same eve temperature may rise to 25°C. We had around 30°C in June which is a bit warmer than average. Winters are not bittercold, but expect rainy and foggy weather then. Just be prepared for everything (t-shirt, sweater, rain-coat etc.)
Photo Equipment: Besançon is not a typical tourist town, so you should be able to get photo equipment for a reasonable price in downtown.
Photo Equipment: It would be a great disappointment to forget your camera after walking to the top of the hill. There are great views from here in both directions - to the west you get an excellent panorama of the town, while to the east you can see the mountains and valley.