We will be arriving at Terminal 3 at Heathrow from Vienna. We are staying at the Luna Simone Hotel in London. What would be the easiest route to get to the hotel? We are not familiar with London at all.
Your hotel is walking distance from London Victoria train station. You can take the Tube from Heathrow to Victoria with one connection. There are several connection points available, and some Forum contributors have advice for which one has the least distance between platforms. See also www.tfl.gov.uk. If you took a faster Heathrow Express train to Paddington station, you'd still need one connection in the city by Tube or taxi, but Tube the whole way is cheaper.
Emma's directions except we always change at Baron's Court. All you have to do is walk six feet to the other side of the platform to pick up the to train to Victoria.