Norlandair (www.norlandair.is) operates flights to/from Akureyri (daily in June; five weekly July and August; three weekly September to May). The 25-minute journey takes in the full length of Eyjafjörður and is an experience in itself. Ticketing is handled by Air Iceland Connect; one-way fares start at around 10,300kr. Flights from Reykjavík to Akureyri can connect to the Grímsey flights. If you book the 'daytour' option (additional €40), instead of the basic airline ticket on the website, it comes with a guided tour.
There is year-round ferry service between Dalvík and Grímsey.
The Sæfari ferry (three hours; adult/child 3500kr/free) departs from Dalvík at 9am Monday, Wednesday, Thursday and Friday, returning from Grímsey at different hours (giving you two to five hours on the island). June to August there is also a service on Sunday.
If you're coming from Akureyri, Strætó's bus 78 won't get you to Dalvík in good time for the ferry. Without your own wheels you'll need to spend the night before in Dalvík.
From Dalvík, and utilising the Sæfari ferry, island outfit Arctic Trip has a day tour (26,000kr), or Dalvík's Arctic Sea Tours organises a 10-hour tour with four guided hours on Grímsey (25,200kr). In Húsavík, Gentle Giants offers rigid inflatable boat (RIB) day trips to Grímsey (72,300kr).