If you want to see Northern Lights, you'll typically need to visit during the winter months and venture outside of Anchorage, away from light pollution... the farther north, the better, as Big4Flyer stated. I recommend following one of the Northern Lights report pages on Facebook to get an idea of when/where you might see them.
If you come in late February/early March, you can enjoy
Fur Rendezvous activities and watch the ceremonial start to the Iditarod in Anchorage. You could even drive to Willow (~80 miles) for the official Iditarod start.
If you want people, shops, and hotels, Anchorage is probably your best bet in the winter. However, it's a lot less quiet both in and outside of Anchorage in the summer months. Girdwood, Seward and Homer are fun little towns to visit in the summer/fall. The drive along Turnagain Arm is beautiful, with scenic views and whale watching opportunities.
Regardless of your destination, a car will be necessary unless you don't plan to venture far from your hotel. Rideshare services are not available and public transportation is spotty within Anchorage.