Geocaching Australia

You can't - not in the city centre. There are lots of places a bit further out, and I hope a Sydney local will come along soon and name some for you. The only one I can think of is Lane Cove. On the up side, Sydney public transport is pretty good. There are trains and buses all over the place and they run fairly often. Pick a train line and follow it out until you come to a caravan park - that's possibly easiest, unless you get a good recommendation from someone here.

I don't think you'd want to drive too close to the city, anyway, if you have a choice. It can get pretty congested, particularly around peak hours (well, for Australia it can, anyway!)

You will have noticed that there aren't any rest areas for larger vehicles to stop at anywhere in or near the city, either. Last time we were in Sydney, we ended up stopping the van in a large carpark next to a closed shopping centre for the night. There was a Mcdonalds with a toilet across the road, so we were OK.

when we first moved to sydney we stayed in a caravan park at Bass Hill that was close to a train line into the city - not a particularly exciting spot to stay though but the public transport is frequent is out that way. http://www.basshilltouristpark.com.au/
Lane Cove is next to the National Park, and probably has a bus line into the city
Sydney Lakeside is in Narrabeen and a fair way north of city centre.

Councils have cracked down lately on backpackers just finding a park in a city street.

Not knowing the area very well, but I would look to the north or west. Gosford maybe, or the Blue Mountains, Katoomba ect. Both have regular commuter train services.

I can't advise on where to stay but I can say Katoomba is well worth the detour.

Not having done it, but Lane Cover River Park is probably a fairly good option. It is probably about 1km walk to the nearest bus/train. The 545 bus would take you to Chatswood where you could easily get a train to the city, and it's a bit closer than the train station there.

www.lcrtp.com.au/

Another website that might be of use on your travels is www.exploroz.com - while a lot of it is set up for 4WD offroad trips it does still have a lot of useful information.