If you're interested in public transportation more conventional than the 4 x 4 bus traversing the interior (or if you're visiting outside of summer, when the former buses don't run), Strætó is responsible for running the bus service along the Ring Road. Most tourists will be interested in Bus 57, which departs daily for Reykjavik. There is also bus 56, which runs eastbound to Egilsstaðir via Mývatn. Finally, there are two spur lines off the main road - Bus 78 to Siglufjörður via Dalvik (and Árskógssandur with connections to the Hrísey ferry), and Bus 79 to Húsavík.
The Ring Road bus stop is a considerable distance from the SBA and Sterna Bus stations on the south side of town. It is actually on Strandgata, just off the Ring Road, just outside the cylindrical official tourist office and about 200 meters from the cruise ship dock.
A fun way to get between Akureyri and Reykjavik is to take a rugged 4 x 4 bus through the interior along the Kjölur route F-35. During the summer (roughly late June - early September), scheduled buses run daily through the interior, stopping along the way for sightseeing at attractions such as Geysir, Gullfoss, and Hveravellir. The scenery is tremendous, and the experience is truly unique.
As of 2013, two bus companies serve the route. SBA Norðurleið has buses that depart both Reykjavik and Akureyri at 8 AM, arriving at the other end around 6:30 PM. The other bus company, Sterna, runs a more staggered and sightseeing-oriented schedule, with northbound buses making an additional stop at Thingvellir. The northbound buses also depart Reykjavik at 8 AM, but don't arrive in Akureyri until 10:55 PM. Southbound buses depart Akureyri at 8:30 AM, arriving in Reykjavik at 8:20 PM. Both buses arrive and depart Akureyri from the parking lot in front of the SBA Norðurleið station (the Sterna office is just across the street). Tickets can be purchased in advance online, or at the bus station if there are still seats available.
The main bus station, run by SBA-Norðurleið, also doubles as an unofficial tourist office. Several different day tour buses stop here (UPDATE: The Ring Road buses now stop on Strandgata, outside the official Tourist Office). In addition, there is daily summertime service between Reykjavík and Akureyri via the Interior. Other summertime buses run from Akureyri to Myvatn and Dettifoss.
The office itself can be useful, with tour booking services, a WC, Internet access, and coffee available.
While it is theoretically possible to get to various sights outside Akureyri with public transportation or day tours, it is much more practical to rent a car. Iceland recognizes US driver's licenses and insurance, though you should check with your insurance and/or credit card company to make sure what is and isn't covered, and whether you should spend extra on the collision damage waiver from the rental company.
All of the major car rental companies have offices in town and/or at the airport. While you can usually walk up and rent a car on a whim, it's usually cheaper to book in advance online. After some comparison shopping, we rented a subcompact from the Europcar affiliate. Since we arrived in Akureyri by bus and were departing by plane, we arranged to pick the car up downtown and drop it off at the airport. The website didn't quite work properly, but a quick Skype call to the office clarified the reservation, plus the agent threw in airport drop off and even a car seat at no extra charge. Note that the airport rental car desks aren't always manned -- if they're not, you can simply drop the keys off in the appropriate drop box next to the Air Iceland check in counter.
Across from the SBA-Norðurleið bus station is the Sterna (formerly the TREX) bus station, on Hafnarstræti 77. This company runs seasonal service through the interior between Akureyri and Reykjavík, as well as selected day tours and occasional service along the Ring Road. However, the regular Ring Road service has been taken over by Strætó, and their buses now depart on Strandgata, not far from the official tourist office and the cruise ship dock.
Akureyri is a popular port of call with cruise ships. Most of them will dock at the eastern end of Strandgata. I don't have specific advice on how to book a cruise here, but follow the web link below for the 2013 schedule of arriving cruise ships.
Akureyri Airport (Code: AEY) is located about 3 km (2 miles) south of the town center. The terminal has a comfortable waiting area, an ATM, and a small cafeteria.
Air Iceland has several daily flights to Reykjavík Municipal Airport (RKV); flight time is about 45 minutes. Air Iceland also provides daily service to the island of Grímsey, as well as the villages of Þórshöfn and Vopnafjörður.
If you wish to fly to Akureyri from outside Iceland, you'll have to change planes in Reykjavik. UPDATE: WOW Airways has taken over Iceland Express, and has discontinued flights between Copenhagen and Akureyri. NOTE: All international flights (except from the Faroes and Greenland) arrive at Keflavík Airport (KEF). As a result, you will normally need to transfer from Keflavík to Reykjavík; allow about 3 hours to do so. During the summer, Air Iceland does provide one daily flight from Keflavík to Akureyri, but the times may not accommodate your particular connection.
There are no buses from the Airport to town. If the weather is good, it is an easy and scenic 35-minute walk north on Drottningabraut to get to town. Otherwise, you'll need to take a taxi (about Kr 800).
This is a schedule for the bus Akureyri Ólafsfjörður.
Here is the one for Myvatn : http://www.english.sba.is/Scheduledbusservice/
There is a taxi station in downtown Akureyri on the west end of Strrandgata. If you aren't on a budget this is a comfortable way to make day tours out of Akureyri. For groups that don't fit into a regular rental car this can be cheaper than renting a car for the day. They offer tours to Mývatn, Goðafoss, Laufás Húsavík and around Akureyri. There are special rates if you order in advance. You can also ask for english or german speaking drivers.