Sarah and Abraham's relief at having a son is obvious and palpable. God has given Sarah laughter, and the oddity yet blessing of a son will cause laughter in those who hear of it. So this gives the duality of sorts in Isaac's name. At first, the laughter was disbelief, surprise or maybe even contempt. But now the laughter is joy or pleasure. What she thought impossible is now realized. When the child is weaned, they hold a great feast in celebration. Clearly Abraham is very pleased.
However, things with Hagar are not fully resolved. Sarah catches Ishmael "mocking". There's a bit of debate about exactly what is going on here, but the NIV translation fills in the blank as "mocking her son Isaac", and I think that's a reasonable interpolation.
Ishmael is, of course, displaced by the birth of Isaac. Isaac is the promised child, and Ishmael is now the discarded first-try and is not loved in the same way by his father Abraham. Also his mother is a slave, and not the honored first wife. So life is perhaps difficult for young Ishmael, who we can calculate to be around 13 years old.
Sarah, at least, is displeased with Ishmael's behavior and demands that he and his mother be driven out. So at last we see the results of Abraham's unfortunate earlier decision. He now has to effectively choose between his two sons, because the strife that began earlier (which, for whatever reason, God chose to reconcile at the time) is proving to be persistent. From Sarah's point of view, she has now borne a son, is no longer in shame or disgrace, and is clearly not interested in having Hagar around anymore. Ishmael is in one sense a competitor with Isaac for Abraham's inheritance, and that is the point that Sarah brings up first. But in another sense, Hagar/Ishmael are a constant reminder of Sarah's earlier barrenness, so that is also possibly an underlying reason for why she wants them removed.
This time things go differently. Instead of God meeting Hagar and telling her to return, God "speaks" to Abraham and tells him to side with Sarah in this matter and remove Hagar/Ishmael. He does so, sending them off into the desert.
Interestingly, while God does not tell Hagar to return, he in fact does meet with her again and provides her a well, a renewable source of water on which she and Ishmael could survive. So he doesn't return her to Abraham, but does provide for her sustenance and as a footnote, Ishmael becomes the father of many nations.
Lastly, this chapter has a bit of political negotiation between Abraham and Abimelech. Abimelech notes Abraham's growing power and demands a non-aggression treaty, in effect. Abraham agrees, but takes the occasion to "complain" about some wells. He's basically asking for some wells to be returned to him. Then he gives him some livestock and they make an oath. Then it is explained this is why the well is called Beersheba. This story is largely paralleled in chapter 26, but I think I will save the discussion for then, because it is relevant to the life of Isaac so I want to have all of the context filled in before discussing the parallels.
And finally, Abraham plants a tree and calls on the name of the LORD, once again demonstrating his piety.