You wouldn't do bad to take the train to either Glasgow or Edinburgh and spend the whole weekend there. You won't get a distillery, but you won't lack for things to see and do, both whisky-related and otherwise. I don't really know Glasgow (hopefully someone else here can advise you on that if you're interested), but in Edinburgh, you could visit the whisky centre near the castle, and taste a wide range of drams in a good pub like the Bow Bar. There are great shops like Royal Mile Whiskies and Cadenhead's, and so far you haven't gotten far off the Royal Mile itself.
If you're planning to hire a car to get around, you might catch a ferry to Arran and make a loop around Argyll. You'll get a distillery or two, great highland scenery, and interesting cultural and archeological sites. If you get into Prestwick early enough, you could do this:
Thursday: Ferry to Arran; if you are early enough, see the distillery. Otherwise, do it Friday morning. There is also a brewery, and a wealth of archeological sites, if that interests you.
Friday: See what you didn't see Thursday, then ferry to Claonaig and drive to Inveraray. Stay at the George Hotel, and visit the amazing Loch Fyne Whiskies across the street. Inveraray is a small town with a few shops along a main street that will take you five minutes to walk down and back if you stop to pet the dog; but it's pretty and charming, LFW is a must, and the George has good food and a great bar.
Saturday: Your choice of loop drives. Start by driving north to have a look at Kilchurn Castle. Then either: 1) head east and do a loop through Glencoe, for the fantastic scenery, then down to Oban (check to see of the distillery does tours on Saturday; not many do). Then back to Inveraray via Kichurn; or 2) drive west directly to Oban, then south through Kilmartin Glen, which is loaded with archeological sites; then back to Inveraray via the shore of either Loch Fyne or Loch Awe. You can probably combine the two loops if you don't mind spending most of the day in the car.
Sunday: Drive to Glasgow. Auchentoshan is on the way, but I doubt there would be tours on Sunday. (Doesn't hurt to check.) Spend the two nights in town. You could do Edinburgh instead--it's only an hour or so further east--if you don't have to be at Prestwick too early. If you do, be sure to leave plenty of time to get to Prestwick--you'll have to go through or around Glasgow, and traffic can be heavy.
Islay is tempting, of course, but it takes a long time to get there, and a long time to get back, and you won't get any distillery tours on the weekend.
Ferry information: http://www.calmac.co.uk
Great general information for all of Scotland: http://www.undiscoveredscotland.co.uk
Good luck, and have fun!