Regular flights (AIC or Eagle Air) connect the Northeast to other Iceland regions and ferries operate to the islands of Hrísey and Grímsey. Long-distance buses have schedules to most towns in the Northeast and the larger towns have public buses - the one in Akureyri is free of charge.
Taxi is nearly always available (in particular in Akureyri) and car pooling is common, see e.g. here.
An overview of public transport (domestic buses, flights and ferries) can be found here.