I could spend a lifetime in Scotland - the bad weather, the dour people, the whisky.
But if you only have 10 days on the ground (assuming 2 travel days on either end), here's a great itinerary.
Fly in and out of Edinburgh (pron: Edin-brah with second syllable faster than first). Glasgow is fine if a better flight, but you will want to see Edinburgh either way so this saves time. Rent a car. You will need to learn to drive a stick while sitting on the right side of the car and driving on the left side of the road. An adjustment, but not beyond you. Just don't hit anyone or anything.
There is much to see and do from a typical tourist standpoint (e.g. museums, castles, shops, and what have you). My favorite things are just driving the incredible roads, hiking, and then repairing to the pub for a wee dram. And for fans of scotch malt whisky, the distilleries are like holy places.
Day 1 - 2
The highlight of Edinburgh is just walking the royal mile that links palace to castle. While touristy as all get out, it's still a lovely and fascinating walk. Explore the little alleys or wynds.
Day 3 - 4
This is my favorite part of Scotland. I could be quite happy waking up every day the rest of my life at the Clachaig Inn. The famed hiking trail The West Highland Way passes through Glencoe. Ben Nevis, the tallest mountain in the U.K. is here, and a popular climb. If you want to do a loop to Skye, you can get the ferry at Mallaig (book ahead) and see two famous sights along the way: the Jacobite aka "Harry Potter train" and Eileen Dolan castle.
Day 5 - 6
Isle of Skye
Over the sea to Skye. A magical island. Be sure to hike the Fairy Pools and the Cullen Hills. Drive the circuit of the island for countless sea views.
Day 7 - 8
The most exciting road in Scotland. Get your Aston Martin on and cross Applecross Pass. The reward after your harrowing journey is the comely Applecross village. Have a delightful dinner gazing back over the sea to Skye and spend the night at the Applecross Inn.
Another true gem. You could spend a week here alone wandering the adorable town hugging the River Tummel, exploring the many ancient standing stones or visiting scotch distilleries including the smallest one in Scotland The Edradour Distillery.
On your way back to Edinburgh and worth a visit. There is the Old Course for golf fans. There is the beach where they filmed Chariots of Fire (cue the Vangelis score), and a lovely little town with a ruined cathedral. This is where Wills met Kate for your royal watchers out there. A good place to sleep before hitting the airport.