Howdy from Burlington.
I lived in Cary and went to Apex high, so I know a little about your area.
For day hangs try Bond park, may be down by the lake. Or lake Crabtree.
Umstead park, Jordan or Falls lakes, raven rock
Lots of other places like Uwharrie and the AT if you head west.
Just do some practice hangs in the back yard before you head out for an overnight. There is a learning curve with hammocks.