By ship
The Rivercat ferry service (http://www.sydneyferries....) runs between Circular Quay to the Charles Street Wharf. Its a very pretty and relaxing way to for a traveller to visit. It takes a fair bit longer and costs twice the price of the train, if you have already "done" the harbour.
At $6.60 for a one way trip, it is an expensive trip or a cheap 30 minute harbour cruise, depending on which way you look at it. Any MyMulti will also cover ferry travel to and from Parramatta although a MyMulti-1 will not cover a return train trip from Parramatta.
By bike
The Parramatta Valley cycleway offers off-road and quiet road cycling to Parramatta from Sydney Olympic Park and Putney. It is quite well signposted, and should be possible to cycle without a map. It has pretty sections following the Parramatta river from Putney and Sydney Olympic Park as far as Rydalmere, before passing over a ridiculously steep footbridge and an industrial areas to the outskirts of Parramatta. It would be quite possible to hire bicycles from Sydney Olympic Park and cycle to Parramatta if you would like a bit of a ride. The bicycle will come in handy to see the sights of Parramatta as well.
Following the route through Putney it is also possible to cycle from the City to Parramatta on off-road or on quiet roads, the circuitous route is nearly 30km, but you miss the a couple of hairy sections around Concord that the more direct route encounters.
Although not as scenic, the Parramatta to Liverpool Railtrail links Parramatta with Liverpool on largely off-road paths.
By car
Parramatta is located next to the M4 motorway Metroad 4. Parramatta is well signposted from the motorway. Surrounding suburbs also have signposted exits from the city.
If you are in no hurry you can use Victoria Rd or Parramatta Rd and avoid the toll. Traffic between Sydney City and Parramatta during peak hours can be extremely slow. Easily twice as long as the time taken for the trip during off-peak times.
By bus
Parramatta is a hub for many buses, which operate from a bus interchange near the station. If you coming from the city, or near a train station, the train will usually be quicker than the bus. Any MyMulti ticket including a MyMulti-1 will cover a bus trip to and from Parramatta although a train trip requires a MyMulti-2.
Metrobus M52 (http://www.sydneybuses.info) operate every 10-20 from Circular Quay via George St and Victoria Road. The trip takes over an hour end-to-end.
The 545 bus (http://www.sydneybuses.info) offers a frequent service to Chatswood via North Ryde, but again, if you are doing the entire trip from Chatswood, probably quicker to get the train.
The North-West T-Way (http://www.t-way.nsw.gov.au/) links Parramatta to Rouse Hill, and the Liverpool-Parramatta T-Way links Parramatta with Liverpool via Bonnyrigg.
Metrobus M92, M91 connect to Hurstville in Southern Sydney and Sutherland.
Metrobus M60 the Hills District and Hornsby.