Hi, sunflower traveler,
I would say that in order to fully explore, you really do need to relocate every two nights, and maybe one three night stay. With seven of you travelling together, it is going to be difficult to find accommodation this late, especially as accommodation is a bit more limited in the Highlands and Islands. However, don't let that discourage you. This is not an insurmountable problem.
You will need a large vehicle - what they call in Britain a "people carrier," but what in the U.S. would be called a minivan. Several of the rental companies offer an eight passenger van. You may wish to travel with as little luggage as you're comfortable with. If you need extra clothing, you can always pick it up in Scotland.
As you are avoiding cities (good!), and you want to see some magnificent scenery, your best bet would be to take the M6 north through the Lake District (you'll be tempted to stop!), and past Carlisle to Gretna, where the M6 becomes the M74. Welcome to Scotland!
There are many ways to go once you have reached Gretna, but as you have eleven days to play with, I'd recommend staying on the M74 until you get to the M8 outside of Glasgow. Head west on the M8 until you reach the Erskine Bridge (M898). Cross the Erskine Bridge and head north on the A82. Stay on the A82 along Loch Lomond, and when you get to Tyndrum, bear west on the A85 for Oban.
That's enough driving for one day! Stay overnight in Oban.
On day two, spend a day on the Isle of Mull. Visit Iona, then take the last ferry back to Oban. Stay overnight in Oban.
On day three, drive north to Fort William, then west on the A830 to Mallaig. Take the ferry to Skye. For hiking, stunning views, and a distillery, Skye is an ideal location. Stay over at least two nights on Skye.
(Normally, I would recommend staying overnight on Mull, then taking the Tobermory to Kilchoan ferry, and driving to Mallaig from Kilchoan. However, it is going to be very difficult to find accommodation for a party of seven on Mull.)
On day five, take the Skye Bridge back to the mainland. Take the back roads to Duirinish and Plockton. If time permits, take a boat trip out of Plockton with Calum's Seal Trips. Take the A890 to Strathcarron, and the A896 through Glen Torridon to Kinlochewe. If your driver has a head for heights, he or she could drive up the Bealach-na-ba, and follow the coast road around to Shieldaig. From Kinlochewe, take the A832 north to Gairloch. Spend the night in Gairloch.
From Gairloch, you have several choices. But I'm looking again at the size of your party. What I'd suggest is to take the A832 to the Corrieshalloch Gorge, then either head east on the A835 for Inverness, or if you need more Highland scenery, go west on the A835 through Ullapool to Ledmore Junction, then the A837/A836/A9 to Inverness. Spend nights six and seven in Inverness.
I like Inverness as a staging area. There are many things to see and do outside the city. There is Culloden and Fort George for history buffs; many good hikes, including Glen Affric and the Caledonian Canal hikes; and a good choice of places to eat, including that Scottish restaurant for the teens.
You may want to spend an extra night in Inverness, and use it as a base for day trips. But if you don't, a short trip down the A9 will take you to Aviemore and the Cairngorms National Park. There are many hikes and lots of activities for everyone. Aviemore can be used as a base for day trips. Two nights there?
Day 10?: Take the A95 to Grantown-on-Spey. If you haven't had enough distillery visits, stay on the A95 to Craigellachie, then the A941 south to Dufftown. From Dufftown, take the A941 to Rhynie, then the A97 to Dinnet. Turn west on the A93, then follow it through Glenshee to Blairgowrie and Perth. Overnight in Perth. If you decide to skip the distilleries, take the A939 south from Grantown-on-Spey to Ballater.