If you are planning on taking day trips out of Sydney via train, Central/Surry Hills are convenient neighbourhoods to find a hotel in. If you are wanting to be steps away from the beach, I recommend finding a hotel in Bondi Beach. It is the most iconic and tourist friendly beach.
Tips on Ubers and Taxis are absolutely not necessary Public transport is highly navigable, frequently running, clean and easy. Don't expect buses to be the disgusting American kind - they are clean and everyone takes them! You can 'tap' on and off public transport with your contactless card or mobile phone. More information here: https://transportnsw.info/tickets-opal/opal/contactless-payments
Good news - the American dollar is strong against AUD. You will rarely need cash, if at all. For reference, I have not used cash in at least 6 months. Tap to pay with your mobile phone is most convenient. If you need to insert your card chip you are easily spotted as a tourist ;)