To synthesize the design, go to the "Processing" menu and select "Start Compilation", or click on the purple arrow icon in the toolbar. Synthesis should be quite fast since the design is small. After compilation is successful, you should have a new .sof file in your Quartus project directory. It should be 703,642 bytes long.
Plug in your DE0-Nano board via the USB connector. Now, go to the "Tools" menu and select "Programmer".
The FPGA is now programmed with your design! (This only programmed the SRAM though, not the onboard EEPROM — so the design is only stored until power is turned off.)
Note: In the future, you can use the command script de0-nano/program.cmd to quickly program the FPGA's SRAM with your .sof file (it uses the Quartus command line programming utility).