Hi! Sounds like you are going to have a wonderful time! As you only have 15 days after Edinburgh, I'd recommend skipping Dumfries and Ayr, as they're in the opposite direction to just about everywhere else that you're going. Unless you have a strong interest in Robert Burns. Paisley, too. It's in what is pretty much an industrial area, and there's not a lot to see, unless you're interested in the history of clothmaking.
As you're committed to the first three days in Edinburgh, I'd recommend picking up your hired car at the airport, not in town. It'll be a lot easier on your nerves! You can take the new high speed tram line out to the airport. You can get a one way drop off, usually at no extra charge, so that you can drop the car off at Glasgow.
I'd recommend going counterclockwise on your tour. You can visit Stirling on either your outward journey or your return journey. As you're going counterclockwise, if you're not doing Stirling first, take the Forth Road Bridge and the M90 to Kinross, then the A91 to St. Andrews.
From St. Andrews, the A91 and A919 will take you to the Tay Bridge, which you'll cross to get to Dundee. If you aren't in a hurry, Discovery Point is worth a visit. You can go aboard the RRS Discovery, which Scott took on his first visit to the Antarctic.
As you leave Dundee, you have two choices: the coast road (A92), which takes you along the North Sea coast, past Carnoustie (golf!!), and through Arbroath, Montrose, and Stonehaven; or the A90 (the fast road). Both take you to Aberdeen.
From Aberdeen, you have two choices (again!). You can take the A93 through Aboyne and Ballater, toward Braemar. Along the way, you can visit Castle Drum and Crathes Castle. At Ballater, turn north on the A939 toward Corgarff. The A939 is the Old Lecht Road, frequently closed in winter due to snowfall. You'll go through the small village of Tomintoul, then meet up with the A95 at Speybridge. Turn left, and follow the A95 to the A9. You'll be about a mile from Aviemore. Once you've left Aviemore, you can follow the A9 straight in to Inverness.
The second option out of Aberdeen is to take the A96 through Elgin, Forres, and Nairn, and along the Moray coast to Inverness. Elgin Cathedral is worth a visit, as is Huntly Castle.
So far, you've made your way to Inverness. If you're going to Skye, there are several ways to get there. The best way to get there would be the A82 south to Invermoriston, then the A887 to the A87, and on past the Five Sisters of Kintail and Eilean Donan Castle to the Skye Bridge. You could also take the A9 north to the A835 at Tore, the A832 from Garve to Achnasheen, and the A890 to the A87 near Kyle of Lochalsh. The latter is the longer, but more scenic, route.
Coming back out of Skye, take the CalMac ferry from Armadale to Mallaig. You'll take the A830 from there to Fort William. Most of the 830 is very scenic. Camusdarrach Beach (Ben Knox's beach in "Local Hero"), near Arisaig, is well worth a side trip. From Fort William, you can take the A82 south to Ballachulish, then the A878 south to the A85, and from there to Oban. If you're interested in visiting the Isle of Mull, you can sail from Oban to Craignure on the CalMac ferry. Alternately, if you're coming back from Mallaig, you can take the A861 south from Lochailort, and catch either the Kilchoan to Tobermory (recommended) or the Lochaline to Fishnish ferries to Mull, and then take the ferry from to Craignure to Oban. Tobermory is delightful!
Once you leave Oban, you can take the A85 to Tyndrum, then the A82 down the west bank of Loch Lomond. The best place to stop along the way would be the village of Luss, near the south end of the loch. Once you leave Luss, the A82 will take you directly to Glasgow. Alternately, if you want to see Glencoe, which seems to be on everyone's "must see" list, you can take the A878 back up to Ballachulish, and turn south on the A82 there.