How did the Persians plan on conquering Greece?

Answer 1

First they bribed the aristocracies of the northern Greek cities to join them.

Then they used their fleet to threaten the southern Greek cities which kept their armies at home to protect themselves against amphibious invasion.

They then planned to defeat the Greek fleet to ensure the cities continued to keep their armies at home and not unite, so the Persian army could pick the cities off one at a time.

The Greek plan was to destroy the Persian fleet o remove the amphibious threat and collapse the Persian sea resupply system. As their fleet destroyed the Persian one at Salamis, the Persian plan went astray, and with no amphibious threat, the Greek cities were able to send out their armies to unite and defeat the Persian army (of which half was sent home as they could not be resupplied) at Plataea.

Two good strategies, the Greek one triumphed because of the sea victory at Salamis. If the Persians had won there, the result would have been reversed.