How do I in code add new field values to a D8 content entity? How do I in code, create a new field value, add it to a content entity, and then have it saved to the DB?

Field API is dead. Long live Entity Field API! clearly shows how to read values, but doesn't mention inserts/updates.
I'm not sure what you mean with "field value", do you mean field items with multiple values?

It's the same as reading them, nothing special:

// Those are identical, not specifying [0] results in that using by default.
$entity-&gt;some_field-&gt;value = 'Whatever';
$entity-&gt;some_field[0]-&gt;value = 'Whatever';

// To add a value for the second item:
$entity-&gt;some_field[1]-&gt;value = 'Something';

// And persist to DB:
$entity-&gt;save();


Edit, the following script works for me and gives me the following output:

&lt;?php
use Drupal\node\Entity\Node;

$node = Node::create(array('type' =&gt; 'article'));
$node-&gt;field_tags-&gt;target_id = 5;
var_dump($node-&gt;field_tags-&gt;getValue());

$node-&gt;field_tags[] = 6;
$node-&gt;field_tags[] = array('target_id' =&gt; 7);
var_dump($node-&gt;field_tags-&gt;getValue());

unset($node-&gt;field_tags[1]);
var_dump($node-&gt;field_tags-&gt;getValue());


Output:

array(1) {
  [0] =&gt;
  array(1) {
    'target_id' =&gt;
    int(5)
  }
}
array(3) {
  [0] =&gt;
  array(1) {
    'target_id' =&gt;
    int(5)
  }
  [1] =&gt;
  array(1) {
    'target_id' =&gt;
    int(6)
  }
  [2] =&gt;
  array(1) {
    'target_id' =&gt;
    int(7)
  }
}
array(2) {
  [0] =&gt;
  array(1) {
    'target_id' =&gt;
    int(5)
  }
  [2] =&gt;
  array(1) {
    'target_id' =&gt;
    int(7)
  }
}


The delta is off then in the last example, but this will probably be fixed in https://www.drupal.org/node/2164601, which might also affect how the code above works..