I'll be in Sydney next week from Monday to Saturday. I'll be doing some tourist stuff, visiting friends, some who are central, others in the suburbs. What is probably going to be my best option for transport, and what ticket options are likely to be suitable for this situation?
Adding a new answer since this needs an update: there's now a new choice in town, namely the Opal card. As of Jan 2015, it covers all public transport in Sydney and nearby, including trains, ferries, buses and light rail.
Opal is your standard pay-per-use 'smartcard', just tap on and tap off every time you get on/off a vehicle. Some noteworthy catches/features:
You can pick up an Opal for "free" at the airport, major stations (not all though!), 7/11s, newsagents, etc, paying only for the value you load onto it (minimum $10, IIRC).
For time being, the MyMulti card covered in Gagravarr's answer still exists, and a weekly MyMulti for the appropriate zone may be a better option depending on your travel patterns, particularly if you'll be transferring a lot. The Opal or Not calculator can help you compare your options. (Disclaimer: I wrote it!) But sometime in 2015, they'll pull the weekly MyMulti too (they've already killed off monthlies and longer), and Opal will be the only game in town.
Good news! After a long time when the only option was daily or weekly MyMulti tickets, Sydney has recently launched a pay-as-you-go smartcard ticketing system called Opal. They have an explanation page on Opal for tourists whick gives quite a good intro.
Similar to the GoCard system in Brisbane, you can't load a daily or weekly season ticket onto an Opal card. Instead, you pay-as-you-go during the day or week, and your maximum spend is capped when it hits a certain cost. Currently, those numbers (for an adult) are maximum $15 in a day Mon-Sat, maximum $2.50 on a Sunday, and a maximum of $60 a Monday-Sunday week.
Because of that, you might find that for some tourist trips (eg just a few days heavy use, in central area week), a MyMulti ticket could still be cheaper, but less flexible.
If you want to go down the MyMulti ticket route, you'd want to buy a week long one for your trip, about the only question is which zones to get. The zones only apply to trains and ferries (all buses covered), so just look up where all your friends live and buy the one that covers it. That said, I'd almost be tempted to say you should get a MyMulti3 no matter what - it's only $17 more than the central zone one / $9 more than the central + ferries one, and lets you visit the blue mountains and the beaches to the south and north.
Otherwise, you could get an Opal card, and pay-as-you-go up to the various caps. You can find the various costs, by mode of transport, on the Opal fares section of their website. You can basically hit the daily pricing cap with a return trip on the Manly-Circular Quay route, so basically any day you head out that way on the ferry all public transport after that is effectively free. Likewise, a trip out to the Blue Mountains with one way on-peak pretty much hits the cap.
The only slight thing to be aware of is that the Opal and MyMulti tickets have a surcharge to use the airport stations, which is quite substantial. If you have a MyMulti or have already hit the Opal cap, and want to avoid paying extra, you can however take the 400 bus from the airport to the nearest non airport line station, which doesn't need the surcharge.