The Arctic Coast Way is a route next to the north-eastern coast of Iceland. You will leave big parts of the ring road behind to see the more secluded parts of Iceland. If you want to see a different part of Iceland and travel in a slow pace, this is the itinerary for you. But of course you can also just choose parts of it to explore.
There are several places on this route you could go for a riding tour or a whale watching tour.
This trip starts and ends "in the middle of the country", so you have to calculate a few extra days for the whole holiday.
It takes about 2,5 hour to drive from Reykjavík to the start of this route. You could still drive that distance together with the other sightseeing spots on day one, but you can also choose to stay the night somewhere on the way to the North. Borgarnes is a great place to find accommodation for example.
At the end of this 9 day route, you'll be in Vopnafjörður. From there you can drive back to the ring road and continue your trip via the east and south to Reykjavík. I would take at least 3 days for this. The other option is to drive back to Reykjavík on the ring road via Akureyri. I would take at least two days for this route.