If you make it to Sault Ste. Marie, you need to visit the locks, of course. There is a viewing stand and interpretative center, downtown, that is free.
There is also a locks boat tour that takes about two hours. It starts below the locks, goes up the American side and returns through the smaller Canadian lock.
In any trip to the Eastern UP, a visit to Taquahmenon Falls is de rigeur. (And just North of Paradise, MI (where the road heads West for the falls), is the old Whitefish Point lighthouse with a museum dedicated to shipwrecks, specializing in the Edmund Fitzgerald.
(You will need to figure your own time constraints to decide whether to take the two hours to go to Tahquamenon from Sault Ste. Marie, adding the time to actually look at the falls and/or add the forty minute round trip drive to Whitefish Point to see the museum (plus the time for the falls and museum, of course).
St. Ignace, at the Northern end of Big Mac, has both tourist traps and actual history, but (no offense to St. Ignace residents) someone with a tight schedule can probably skip it.
Mackinac Island, (The second National Park in the U.S., although the Feds returned it to the state of Michigan), is wonderful, but a day hardly does it justice. I’d recommend making a trip to see it, alone, later. (No private motor vehicles are allowed on the island, so if you bike it is a great place to explore.) You can do the island (village, fort, shops, and scenery) in one very full day, but it is a bit rushed. If you are set on visiting the island, the ferries are run by Shepler, Arnold, and Star Line. (Rooms at the Grand Hotel are based on prestige, with tiny rooms costing a fortune. The hotels in “town” are cheaper (not cheap). (You do not want a room facing the harbor: the sun will bake the tiny room and the building across the street will block the view of the lake.) The B&Bs are probably nice–and expensive. If you are not going to spend more than a day on the island, consider sleeping on the “mainland” and taking early and late boats to get on and off the island.
On the south end of Big Mac is Mackinaw City (village). It is even more touristy than St. Ignace, but it does have a good replica of Fort Michilimackinac that shares space with an historic lighthouse and archaeological information of shipbuilding. (All the parks in the area have a joint web site at Mackinac State Historic Parks. Mackinac and Mackinaw are both pronounced “MACK in aw”.
Cheboygan and Alpena has some historic stuff, but since you seem to be tilting Westward (where there is more stuff), I will concentrate on that side.
Take US-31 South from Mackinaw City to County road C-66 toward Levering and Cross Village. In Cross Village, pick up M-119, South, and follow it along the lake shore for a psychedelic, (and occasionally hair-raising) twisty trail with intermittent views of Lake Michigan through beech/birch/pine forest. (If your ball joints are not 100% reliable or you are concerned about highways that are uncharacteristically (for Michigan) shoulderless, skip this excursion.)
At the South end of M-119, the village of Harbor Springs has some beautiful 19th century mansions. Following the shore of Lake Michigan, you pass through Petoskey, Charlevoix, and Traverse City. From Traverse, M-72 takes you to Empire and the Sleeping Bear Sand Dunes. (Local/County web site.) (Take the time to study the maps and pick a good place to cross the dunes to Lake Michigan. The old “traditional” location for starting a dunes hike (on M-109, North of Empire, at the West end of Glen Lake) began with an 800+ foot high trudge up shifting sands, only to discover at the top that you were still one and a half miles from the beach, (to which you had to hike down, then back up to return.) (There are now other, shorter, park trails through lower dunes and forests for those with less stamina or less spirit of adventure. You’ll need to pick up a decent map from the NPS.)
That should take care of about six of the three days you have allotted.
For future (longer) vacations, there are a myriad of cabins throughout the whole Lake Michigan shore region, at least as far South as Muskegon and as far North as Petoskey. All the Boynes are in the Petoskey neighborhood, and there are a number of placid rivers with canoe rentals in the area. (Michigan has several challenging canoe rivers, but I do not believe that any of them are in that area.)
Another trip that is too long for your current vacation, but which you may want to keep in mind for the same region is a tour of the Hemingway sites, in which the locations of the “Nick Adams” stories (written using locations where Hemingway had lived and toured) are highlighted for touring.