Hi Marana and welcome here! The easiest way how to check the CX1 shutter count is to switch the mode dial to Scene mode, then while pressing the ADJ./OK button to DOWN direction, press and hold the Playback button for a second or two. In the appeared list of numbers check the "SH" line. This number represents the shutter count.
Thank you so much for this information, it works like a charm! My shutter count reads 1254 already, and I've only had it less than two weeks, guess I'm in love, :^) Do you know what the other info means?