Grand Marais is a good jumping off spot and there is much to see and do doing daytrips from there.
Definitely go up the Gunflint Trail to the end.
Have lunch or dinner at Gunflint Lodge.
Go to Chik-Wauk Museum and learn about the history of the area (it is near the top end of the Gunflint Trail).
Go have dinner at Nanaboujou Lodge, beautiful place.
Be sure to visit High Falls at Grand Portage State Park.
Visit Grand Portage National Monument (fur trading history).
Cross over into Canada at the border just to do it!
Cascade River State Park (south of Grand Marais) has a great trail that follows the cascading river and pools.
Grand Portage Indian Reservation with its history. Also a casino there, of course.
But, there are great places that are nearer to Burlington Bay.
Gooseberry Falls State Park has a great visitor center and beautiful waterfalls.
Split Rock State Park and Historical Site has an iconic Lighthouse that you can tour, and a great visitor center too.
There is a paved bike trail that connects the two state parks and goes farther up the shore.
Hike the Superior Hiking Trail that offers great views of the lake.
Where to stay...your choice. Either way, you won't be without lots of things to see and do.