The big tour operators have "Lakes and Mountains" programmes, so you could start there for ideas. The advantage for the traveller is that everything is organised and taken care of. The disadvantage is that these holidays are only in a few specific locations, using specific hotels and departing on specific days, ruling out any flexibility. If you are confident enough to book everything yourself, you can put together an itinerary that better suits your needs, dates and budget.
Self booking Austria is very easy. There are plenty of flights from the UK to Salzburg (BA, Easyjet, Ryanair, WizzAir, Jet 2) and accommodation can be booked either through booking.com/airbnb/vrbo, through the local tourist offices or directly with the hotels, B&Bs (called "Pensions") and apartments.
Salzkammergut, east of Salzburg, is lovely. Plenty of lakes like Wolfgangsee and small, atmospheric towns and villages. West of Salzburg takes you into the mountains. Zell am See/Saalbach/Hinterglemm are all popular destinations with access to lakes, mountain hikes, biking, zip wires, paragliding and much more.
You could also consider driving. UK-Austria by car is a trip we do regularly. For the round trip we reckon about £300 for fuel, plus £100 each way for the ferry (Dover - Dunkirk) and £100 a night for an overnight stop in Germany each way at a IBIS/Travelodge type of hotel - so in round numbers, the travel element costs about £700. The journey from Dunkirk takes about 12-14 hours but we prefer to split the journey.