Can I travel to Schengen countries with a Czech Republic employee card?
I think you are actually asking if you can use a Czechia government employee card as a form of government-issued identification to travel within the Schengen area.
Typically, you are required to have a national identity card from a Schengen country to travel within the Schengen area, or if you are not a Schengen-area citizen, a passport from the country of your citizenship. While using a Czechia government employee card may be decent evidence of a right to live in Czechia, it is unlikely that other countries will recognize it and you will be taking unnecessary risks trying to travel with it.
Travel with your Schengen country ID card, if you are a citizen of such, otherwise travel with your passport. (A passport is always OK, even if you are a Schengen area citizen.)
If you have a legal permission to stay and work (tourist or national visa) in any Schengen country like Czech Republic you can travel to any other country of Schengen zone by air or land transport.