Easiest way to get out of the hustle of the city is to go to one of the beautiful beaches around Sydney. The 3 most popular would be Bondi Beach, Manly Beach and Coogee Beach.
Cheapest way to get there (next to walking four hours, would be the bus. From the city you have got the following bus routs to the beaches:
Bondi: Rout 333, 380, 381 and 382
Coogee: Route 372, 373 and 374
To get to Manly you need to get onto the ferry from the Circular Quay. The ferry in waft 3 will get you to Manly. (By the way - that is also a very spectacular way to see the Sydney Opera House!)
Next to swimming and sun bathing there are some beautiful walks around the beaches. Coogee to Bondi walk which will take about 2.5 to 3 hours, if you don't feel like walking that much, take the bus (route 378) to Bronte and walk from there to Bondi, which will only take you about an hour.
On the north side you can get a bus to the north shore and walk the Spit to Manly walk, which is about 10km and 3-4 hours long. Bus routes to the Spit Bridge are route 190, 169, 178,179 and 180. Those busses will also bring you further out towards the Northern beaches and Palm Beach (where Home and Away is filmed).
The famous area of the Blue Mountains is a popular day trip - but you can easily spend a couple of days up there if you'd feel like it. Easiest and probably cheapest way to get there is by train. From Sydney take the train to Katoomba, which is also a commuter train and runs quite frequent. It takes about 2 hours to get there and the train stops at Blue Mountains towns of Wentworth Falls, Leura, Medlow Bath, Blackheath and Mt Victoria. As there is so much to do around the Blue Mountains you best check out the NSW Tourism website
If you don't feel like organising the trip by yourself, there are many coach companies which offer day trips to the Blue Mountains from Sydney, most leaving from Circular Quay.
Coach tours can be very different in price and are worth comparing - though often the most budget version of the day trip is organised by OZTRAILS and price starts at about $80 for a daytrip. More Luxury personalised tour would be with companies like Blue Diamond Tours and cost about $160 for the day trip. Then again, often you pay for what you get - one include breakfast and lunch, others don't.
An other option for a day out (or two or three) is the Hunter Valley. The hunter valley does not only offer the Royal National Park, but is also Sydneys foody region with wine, breweries and chees factories. If you don't rent a car by yourself you are probably best off to go on a tour. Worth your money might be the Kangarrific Tours guided tour witch offer morning tea with friendly kangaroos, wine chocolate, cheese & beer tasting with only small groups and new buses.
If you would rather want to experience the Royal National Park, why not check out the Understand Downunder Tour which do not only offer picnic lunk in the park with spectacular views, but also offer guided bushwalks.