If you are coming from Bandar Abbas the simplest way is to get on a speedboat at main harbor. There are lots of them departing each few minutes. The ride takes between 30 to 60 minutes depending on the boat. A car ferry leaves regularly from Bander-e-Pol, about 60km west of Bandar Abbas. It is allowed to go on the island with a foreign vehicle (but it’s necessary to show your Carnet de passage).
If you are traveling from other cities, then it is easier to book a direct bus to Qeshm instead of going to Bandar Abbas and then take the ferry. Note that the buses coming from the west don't take the ferry from Bandar Abbas to Qeshm but cross more west towards Laft. Also, the buses might drop you off near Dargahan. Unlike Kish Island you need a visa for Iran if you are arriving from another country; you don't get it on arrival.
There are flights from major cities to Qeshm Island by different airlines.
There is a daily flight from Dubai that costs about US$ 40.