For political, cultural, and historical reasons. In modern historiography, the focus has increasingly shifted towards documenting cultural and political contexts which help to rationalize the decisions of historical actors, instead of assuming a theoretical optimum.

Alexander was bound by the concerns of his political constituency, namely the former Greek city states. He was not the king of Greeks, but was named the Hegemon, and leader of a Greek alliance that aimed to defeat Persia. Greek alliance to defeat Persia was a common tool of legitimacy that the hegemonic contenders the Aegean - Athens, Sparta, and Thebes - used to legitimize their authority long before the arrival of Macedonians. It was arguably created by the Athenians, who, in establishing and controlling the common treasury of the Delian League, and appointing themselves its military plenipotentiaries, created the motif of legitimate rule via destruction of Persia. Sparta, after gaining dominance against Athens served the role of the chief antagonist to Persia, embarking on expeditions to Sardis and Egypt, with no long-term success.

Macedon was the fourth in a line of hegemonic contenders who claimed hostility to Persia to be their legitimacy in ruling Greece. Ever since the Persian invasion of Ionia and Greece, Persia served as the villain in Greek politics. Alexander's invasion of Persia would increase his credibility in Greece, and was therefore a political move.

Some historians have asserted that Alexander was also partially motivated by his tutelage under Aristotle and other famous Greek philosophies and sophists, who regarded conquest of the East as the final accomplishment of Macedonian kingship. There is less evidence to support this view, but it does bring up another point - the epistemology of the ancient world. Unlike the modern world, where geographic education is often imparted in distances and expanses, the ancient world often educated its elites in geography through directions and waypoints. For example, a Macedonian nobleman would not learn that Sardis was x miles across the sea from Athens, but that to get to Sardis, one had to go due northeast at 23 degree angle from Athens, and to get to Tyre from there, one would have to march through the Anatolian passes from Gordion. This made the ancient world "smaller" in the eyes of the educated elites, and made conquest seem all the more feasible.

Alexander in some ways was pre-set in his intention to conquer Persia since it had been the ambition of his father, and preparations were already made. Philip had made the conquest of at least the Western coast of Persia the partial basis for his legitimacy in Greece, and his constituents expected him to follow through.

Most importantly, Alexander was made aware of the military weakness of Persia compared to the Greek hoplites and Macedonian Phalanx by the experience of Greek mercenaries in Persia. When Cyrus the Younger fought for the Persian throne, he enlisted the "ten thousand", a force of Greek mercenaries. On Cyrus's death, the mercenaries were forced to fight their way back to Greece. Surprisingly, they succeeded, demonstrating the superiority of Greek tactics. Xenophon was one of the commanders, and left a lucid and large historical record of the experience.

Conquering Persia was therefore not some heroic exploit or fool's errand that was made right by brilliance. Rather, it was a sensible political and military move. Like Germany invading Russia in 1941, Alexander's more successful invasion of Persia was rooted in a belief in operational and tactical superiority that would offset numerical differences, and the need to unify his own people behind an agreeable common enemy.

As for his plans after Persia, it's almost certain he had none. Alexander never even intended to conquer the whole of Persia during his initial campaigns, regardless of the hindsight-influenced claims of his contemporaries. Alexander's motives for invading Persia and the uncertainty over his initial battles suggest that a simple elimination of the Persian capability to threaten Greece would have sufficed to achieve Alexander's political goals. Indeed, Alexander initially focused on conquering Persian seaports along the Mediterranean, in order to destroy the Persian fleet's ability to base. In theory, it made more sense to punch through to the Persian capital if the intention was to conquer Persia - this suggests that the campaign's original intention was to secure Greece rather than usurp the Persian throne. After Issus, Tyre, and Egypt, Alexander appeared emboldened enough to take on the Persians in a decisive battle in Gaugemala - his adventure in the Levantine Ports and Egypt had given them time to assemble another army. He could very well have been doomed here, but in this climactic battle, Alexander secured mastery of Asia.

After the conquest of Persia, Alexander disciplined rebels in Central Asia, before continuing onto India. The Indian opportunity made sense. Alexander was bolstered by new Asian reserves, and India was divided. The oft-cited, probably exaggerated figure of "750,000" troops opposing him is most likely untrue. India at the time was divided into warring states, and conquest was possible. However, Alexander throughout his career had cultivated in himself a complete ignorance of harsh conditions, a hardness not shared by his men, who threatened mutiny after a decade of campaigning if they could not go home. This terminated his Indian ambitions. Alexander returned to Babylon, and planned an expedition to Arabia, but his health declined with the death of his lover Hephaestion, and he died at 32, unable to execute his vision.

Sources:

Peterson, Robert. A Study of Combined Arms Warfare by Alexander.

Tarn, W. W. Alexander: the Conquest of Persia.

Boardman, Andrew. An analysis of the generalship of Alexander 111 of Macedon: undermining or underlining greatness?

Eckstein, Arthur. Contesting the Greatness of Alexander the Great.

or would he have gone straight to China?

Just to add to u/Interpine's reply: the accounts of Alexander's campaign suggest that he wasn't aware of China's existence. Greek geographical knowledge stopped at India, and as Alexander advanced further into that subcontinent, he fully expected to eventually encounter the outer sea that ringed the continents of the world. This was part of the reason why the mutiny of his army was so frustrating to him: he thought he was nearly there, and that when he reached the sea he would have conquered all of Asia. His alleged plan upon his return to Babylon was to conquer the Arabian peninsula, and eventually turn west against Carthage.

I'm sure more can be said, but there's an excellent answer by /u/iphikrates on a similar question from a few months back - hopefully that helps!