Akureyi is a small city, so walking or biking is the best way to explore it. If you prefer, there are also buses available to take you to different parts of the city. A single ticket is 350 ISK, you can find them at supermarkets or kiosks. Travel time varies depending on the destination, but most places in Akureyi are within a few minutes' walk from each other.
If you have more time, consider taking a day trip to the nearby Lake Mývatn geothermal area or go whale watching. If you have less time, you can skip the Arctic Botanical Gardens to save time and money. Also, don't forget to dress in layers and bring comfortable walking shoes. Akureyi can get chilly, even in the summer.