In the circuit diagram below do I connect to two TUNE1 points together? Or am I misinterpreting what this means?
Yes. The arrow indicates a schematics connector, and the name next to it (in some EDA packages inside it) is the net name. A net is the collection of all pins and signals which should be connected together.
If you would move from schematic to PCB layout you would start with your components placed randomly and the nets represented as a rats nest, a bunch of lines showing all required connections.
(click image to view full size)
As you route the PCB more and more lines will be replaced by copper connections. If you wouldn't connect both TUNE1 pins the PCB layout program will show a line between pin 3 of the IC and R2 after all other routing is done.
If you find routing a PCB an interesting challenge you may also enjoy Planarity :-)