While a Sim is active, simply click any object, and a pie menu appears that shows all the possible actions the Sim can take. The basic way to getting around the world is to interact with objects and other Sims.
The Sims 4 does not feature an autosave! After you move in, you should immediately hit Escape and save your game, and routinely save every so often! Sims 4 is much more stable than some of the previous iterations, but saving early and saving often will prevent terrible outcomes.