You cannot send a probe to the Moon or Mars until you have completed the specific "Mission to [Planet]" research project. Once researched, select the planet and use the launch button at the bottom center of the screen.
For a deeper look at navigating the complex UI and mastering the 100-hour-plus tutorial, check out this guide: Terra.Invicta.v0.3.35.Early.Access.zip
Hire at least two councilors with high Persuasion scores. Your immediate goal is to win over small nations quickly to build influence before tackling global powers. You cannot send a probe to the Moon