there's a farily easy to get trechnology in the "governance" tech tree. after xeno commerce, you can pick one of 3 commerce specializations. the "supportive population" gives you a flat +4 approval per planet. that alone should immediately get you into the "happy" zone - the colony capital provides a +3 base approval and witht the +4 from the tech, you can have tp to 7 billion people at 100 approval, or about 10 million people at a reasonably happy level.
after the commerce specialization, the tree branches into lines, one of which (xeno entertainment and the following techs) gives you access to approval buildings and approval starbase modules. some techs later in that branch, there's also another opportunity to pick up another +4 passive approval.
if you're playing the tutorial, there's an approval relic not far from your starting planet. you just have to build a contructor, use it to build a starbase near that relic and add the xeno archeaology lab (or whatever the module is called). approval relics improve the morale empirewide.
