Comfortable walking clothes - they don't need to be high-end hiking stuff if you don't otherwise do much walking, but should be comfortable. Shorts, long trousers, jeans all suffice. You'll need a lightweight, windproof and showerproof jacket. Layers are good, so t-shirts/tops, a light fleece for cool days (or for when you're at the top of the mountains) and for evenings sitting out at a cafe or restaurant.
Good walking shoes or walking boots are essential, as are suitable socks. You'll need a small rucksack (16L size is ample - e.g. Fjällräven Kånken or similar size) for carrying an extra layer/the jacket and for drinks and snacks and suncream and Antisan insect bite creme. OTC painkillers are very expensive in Austria, so pack some Ibuprofen, Paracetamol or Aspirin in case of aches, headaches and hangovers. A hat, neck-warmer/Buff and sunglasses are also usually in my rucksack.
The weather can change very quickly in the Alps, so be sure to keep an eye on the forecasts. Afternoon and early evening showers are common even on very hot days. Before your departure date you can look at Bergfex for a fairly accurate 10-day forecast. Bergfex is also useful once in Austria, along with the usual apps on your phone.
Unless you're at a 5-star hotel, there is no dress code in Austria, and certainly not in the mountain resorts, where smart-casual, or even casual-casual prevails. I live in the mountains and live in shorts/t-shirt top/sweatshirt from May to October.