The city is accessible by bus (or car) via Route 1-Ring Road, the highway that circles the island along the coastline. It can also be reached more quickly by the smallish airplanes which connect Icelandic cities, flying over the deserted/partially-glaciated interior.
Being Iceland's northern capital city doesn't mean it's a huge metropolis. Although it can get a bit hilly leaving the waterline, walking distances are reasonable for a fit traveller. The public bus system is free for all.
By thumb or foot
Iceland is typically a great country for hitchhiking, but be extremely careful of the weather in this region, as it is highly unpredictable. As late as March or April, terrible Arctic blizzards can blow in off the sea, and hit you in an instant. If you're even a couple kilometers out of town, and one of these storms hit, you will be stranded for an indefinite amount of time. It's wholly possible to freeze to death, or at least come down with hypothermia or frostbite, in such a situation.
Akureyri is a good base to explore some of the more remote areas of Iceland, such as Myvatn. Bus tours are available which take you to see the sulfur fields of the Devil's Kitchen, the enormous crater of Viti, and Godafoss (waterfall of the gods).
Typical of Icelandic cities, there is a heated outdoor pool and hot-pots available to the public. Thorough showering before and after bathing or swimming is expected.
A local way to eat a hamburger is with french fries included in the bun and kokteilsósa (a cocktail of mayonnaise and ketchup). A shop that sells a delicious version of this called a MacGratsky burger is Nætursalan located downtown right next to the main bus stop. This is a popular snack to end the evening after a heavy night of drinking. Akureyri also has the world's northernmost Domino's Pizza. Rub23, located just below the cathedral of Akureyri, has a most friendly atomosphere and serves one of the best fish dishes in the area.
The Indian Curry Hut, located at Hafnarstraeti 100b, has some of the best curry in all of Iceland--perfect for warming up after a winter's day! It's a take-away, though there are two stools and a counter. The owner of the restaurant is very friendly, with an interesting life's story.
Strikid: Fresh, high quality restaurant. Their chocolate cake is the best I've ever had, hands down. Mostly fish, lobster, but one or two meat dishes. Has an amazing view at the top floor of a building at the coastline. Greifinn: Considered high quality, just sells sandwhiches, starters, soups. Obviously a nice restaurant though. Frederick the 5th: Haven't been, but apparently is a "Slow Food" restaurant, where it takes about 3+ hours for all of your courses to be finished. Supposedly a great experience, extremely high class. The chef cooks European cuisine, but uses local ingredients so adds an Icelandic touch.
A nice bar is across the street from the main movie threater. It seems that people in the city enjoy an 'early' movie theater around 8pm or so, and, as customary in Iceland, go clubbing/barhopping after midnight. However, the clubs close at 1AM, at least during the weekdays. A strange wheel-of-fortune with shots, beer, and nothing, is there, for 1500 Kr.
There are many guesthouses more inland immediately away from the city center. They double as residences for college students, but they are the cheapest lodging you'll find, at around 8000 Kr double and 5000 single.
Several buses run mostly hourly from the city centre to other important places like the airport and neighbouring settlements.