Bring some good waterproof mid or high hiking boots with you. The walking paths here are rocky and technical because we are in a high alpine region. Mid and high walking boots give you more stability as a low walking/trail running shoe and will protect your ankle joints. A hike normally takes several hours and a hiking boot has more support and cushioning as a trail running shoe and is therefor in general more comfortable during hikes.
Best you go to an outdoor shop and try different brands and sizes because the custom fit differs quite a lot between different brands.
The weather in the Alps can change very quickly. Depending on the weather you can hike in shorts and a t-shirt or top. In general always bring a wind or rain jacket with you, an extra layer and eventually a rain trousers. It can be colder the higher you get and especially on exposed areas like mountain passes or ridges the wind can be a lot stronger. During breaks you are cooling down quickly and you will be happy with your extra layer or jacket. Sometimes it is necessary to bring some gloves and a hat too or some thermal clothing depending on the weather forecast.
A backpack to put your food, water and extra clothing in. Personally I always put a rescue blanket and First Aid kit in it as well. The same applies to backpacks as to hiking boots. There are different brands and sizes and you better try them to see what fits you well. Well known brands for backpacks are: Osprey, Berghaus, Deuter, Vaude, Millet, Low Alpine
When you decide to sign up for a hike I will send you a specific material list with the equipment you need to bring with you.