Staying a week, suggest you choose self contained accomodation near George Street on the Light Rail Line, Get an OPAL Card, so you can just hop on and off. Paddy's Market has reasonable fresh fruit and vegetables. Lots of places have special reduced cost meals at lunch time, like Chinatown. Sydney doesn't have to be expensive. I always catch the ferry to Watson's Bay, where I have fish and chips and salad on the jetty. I also get the ferry to Manly, where there are a lot of inexpensive places to enjoy lunch, before you walk along the Esplanade to Shelly Beach.
Sydneyinfo.net.au
Light Rail Line Map