Alexander III of Macedon went east and south in his conquests because of few reasons. The Greeks were already conquered by his father, but he needed to secure his authority there so he made a few big blows for them, especially because the Greeks despised the fact that the Macedonians conquered them.
The most advanced civilizations were on the east, and also they were the most powerful, with the Persian Empire being at the top at that time. Knowing that in order to be able to create a large empire he needs to eliminate the main threat, Alexander went for the Persians and managed to win.
In the south, Egypt was the traditional place of civilization, prosperity, and it had a well established reputation, so Alexander went for it and took it under his control.
He had plans to go westwards towards Rome, but since Rome was nothing special at that time, neither in cultural manner nor in militaristic one, he went for what was more important.
