Why did Avraham give Hagar only bread and water, but no money?
Rashi: Because he now hated him for having strayed from the right path. 1
Ramban (on Pasuk 15): Because Sarah ordered him to do so, and Hashem instructed him to comply. 2
Chizkuni: Avraham and Yishmael were converts. Kidushin 17b says that converts do not inherit each other, even mid'Rabanan. Chachamim are displeased if one returns a loan to a convert, the son of the lender [lest people say that they are Halachically related]. Therefore, Avraham did not want to give to him. However, he intended to give to him gifts after Sarah died, like he gave to "Bnei ha'Pilagshim."
Rabbeinu Bechayei: Through prophecy, Avraham knew that Yishmael's descendants will oppress Yisrael. "Im Ra'ev Sona'acha Ha'achilehu Lechem v'Im Tzamei Hashkehu Mayim" (Mishlei 25:21).
Kol Eliyahu: A Midrash learns from here that Avraham's house were Vatranim (very generous). This is astounding! Rather, Da'as Torah is that eating is more than drinking (Megilah 12a). It says in Avos (6:4) to eat [as much as you want of bland] bread with salt, and drink water bi'Meshurah (a tiny measure). Normally, one drinks in a Kad (small flask). Avraham gave Chemas Mayim (a large pouch).
Redak: He also gave money, just the verse did not mention it. Surely he did not send them empty!
Ohr Yechezkel Darchei ha'Avodah p.174: Even though Yishmael was sick, now Avraham needed to use the Midah of cruelty.
It is difficult to understand the difference in Avraham's reaction towards Yishmael and his reaction to S'dom (who did not stray any less than Yishmael). Refer to 21:14:1:2*.
Which explains why he did this in spite of the fact that it went against every fiber of his being (the greatest Ba'al-Chesed who ever lived).
What are the implications of the word "Vayeshalchehah"?
Seforno: 'And he accompanied her. 1
Targum Onkelos: 'And he sent her'.
Targum Yonasan: 'And he gave her a Get'. 2
Why did Avraham place Yishmael (who was at least 16 years old) on Hagar's shoulders?
Rashi: Because Sarah gave him an Ayin ha'Ra and he became feverish, unable to walk by himself.
Ramban: The words "ve'Es ha'Yeled" belongs earlier, after "El Hagar". In other words, Avraham put bread and flask of water on her shoulder 1 , and gave to her Yishmael, to accompany her wherever she went.
Why does the Pasuk see fit to add that Avraham placed the bread and water on Hagar's shoulders?
Targum Yonasan: It hints that he also tied them 1 to her loins, as a sign that she was a slave.
What does the Torah mean when it says that she strayed in the desert?
Rashi #1: It means that she reverted to the idolatry of her father's house.
Rashi #2 (on 20:13), Rashbam, Targum Onkelos and Targum Yonasan: She literally lost her way and strayed in the desert of Be'er Sheva.