Why did he take a stick and stones?
Malbim: David trusted so much in Hashem, he did not take any weapons, only what one uses to expel beasts. The blasphemer was like a beast in his eyes!
Why did he take five stones?
Radak (citing Midrash Shmuel 21:5): One was l'Shem Hashem, one was l'Shem Aharon, and three l'Shem the Avos. Hashem said, I should 'pay' (punish) him - he blasphemed Me! Aharon said, I should pay him - I am Go'el ha'Dam 1 . The Avos said, said, we should pay him - he reviled "Ma'archos Elokim Chayim", he wanted to uproot our children, and to uproot their Torah, which is their sustenance.
Radak citing a Midrash: They are five for whom David had Kinah (zeal to avenge the insult to them) - the five Chumashim, Hashem's name and the memory of three greats of the world - Avraham, Yitzchak and Yakov.
What are "Chalukei Avanim"?
Rashi: They are small, smooth stones.
Radak citing Targum Yonasan: They are smooth stones, proper for a slingshot.
What is Yalkut?
Targum Yonasan: It is a Tarmil. Radak - this is a Kli of leather or cloth in which travelers put their food for the road.
Radak: It is a Kli in which a slingshot shooter puts the stones that he is Loket to put in the slingshot. This is why it is called Yalkut!
Why did he take both a shepherd's Kli and a Yalkut? Surely one could hold all five stones!
Radak: Perhaps he had more stones in one Kli, and the five that he chose in the other Kli.