MiSTer project that is installed on the DE10-Nano board can only output video using the on-board HDMI connector. However if you want to get the real retro gaming experience then you can attach the MiSTer IO Board. Among other things, this add-on board let's you output 15KHz analog RGB video and audio signals that will let you connect to a CRT monitor. This blog post describes how to build such a board.
In order to connect your DE10-Nano board to VGA or other RGB output, you need an I/O Board. This PCB board and assembly instructions are available on the GitHub pages of the MiSTer project.
Getting FPGA cores of your favourite retro systems onto the DE10-Nano is very easy. You can run them in matter of minutes.