Brewster, Mass.

If you can't get into Nickerson SP there's Sweetwater Forest that's close by. We've stayed there several times over the years.
We've gone on the Cape using Walt's directions. Avoid arriving on a Friday night or leaving Sunday evening, otherwise you'll run into all the weekend traffic.
Tale your bikes - the Cape Cod Rail Trail is a nice, easy to peddle multi-use trail.

I am assuming you are coming in from MI.

Head into the state using RT 90 (Mass Pike - Toll Road) heading East to RT495 South - Stay on 495 South down to 25 down to the RT 6- Heading over to the Sagamore Bridge - This will put you on to RT6 on the Cape. Route 6 all the way down to Exit 12 and that will head you into RT6a and heading into the State Park.

None of this will bring you near Boston and small streets/ways. That is not to say you will not have traffic - most of the summer the Bourne and Sagamore Bridges are very congested.

Nickerson is a nice state park - dry camp only last time I was there a while ago and small/medium size sites and roads. Generators are ok but even if one person complains they will make you shut it down. (just fyi)

Maybe you could stay at Nickerson State Park. On your way you could stop at Nickerson Family Campground in Chaplin CT. I have never been there so I don't know if it's any good or not.