Hi mate,
Have a look at this
http://www.dartmoor.gov.uk/__data/assets/pdf_file/0007/43909/vi-camping_map_309.pdf - if the link does not work, just google 'Dartmoor wild camping map'. This shows you were it is ok to camp.
It depends on how long you plan on going for, how far in you want to hike, and where you want to leave your car. For example, you can leave your car near Fenworthy reservoir, hike in through the forest and into the moor enough to cross the stone wall and into the wild camping zone. There is a great camping spot just over the wall by the North Teign River which my wife and I discovered last easter. It would not take you more than a few hours to walk in.
Alternatively, if you want to take a few days then start from Belstone in the North and walk in (can leave the car in Belstone village).
Please be careful, do some research (google is good!) - make sure you bring an OS map and compass and/or GPS as it is not hard to get lost, particularly if it gets foggy. Plan your route and let someone know where you are going and when you will be back so they can alert the authorities if you are late. Take the right gear depending on the weather, and make sure you leave the car somewhere safe and with no valuables visible. Bel stone would be safer than Fenworthy I reckon.
Sorry, I have no experience with the south downs if you are looking to go there.
T