By plane
Royal Jordanian operates 2 daily flights between Amman and Aqaba, one in the morning and one in the evening. Duration of the flight is approximately 1 hour and costs 37 JD + tax one-way.
The King Hussein International Airport is located north of Aqaba, about a 20-minute drive.
By ship
Ferries run regularly from Aqaba across to Nuweiba on Egypt's Sinai peninsula, bypassing Israel and the sometimes complicated border arrangements. Generally there is no visa fee for entering Jordan through Aqaba since it is a part of the free trade zone. The line to Nuweiba is operated by ABMaritime, see their website for the 'official' timetable and current prices.
There is more information about the boat crossing in the itinerary Ferries in The Red Sea
By Bus Or Car
The Desert Highway terminates in Aqaba. There are frequent buses to Amman and other points along the highway. Fare is currently 7 JD each way.
Amman to Aqaba is about 350km using the Desert Highway. It will take about 4 hours to travel this distance at a reasonable speed. Do remember that service/fuel stops are not very frequent on this road.
The Israeli border at Arava for Eilat is only a short hop away. The Saudi Arabian checkpoint is also visible on a clear day, but visitors without a visa can only look.
If you are planning to come to Aqaba directly from northern Israel e.g. from Tel Aviv, Jerusalem, Be'er Sheva, etc. by catching an Egged bus to Eilat, you should ask the driver to let you off at the "Eilot stop to Jordan", which is the last stop before the Egged station in Eilat, and then walk the <1km to the border checkpoint.