First of all, I'm assuming Tioga Pass will be open when you are making this trip. Last year it opened June 24th, and this year's opening will also be late.
Driving from Furnace Creek to Lone Pine will take about an hour and a half. Lone Pine has a few restaurants, MacDonald's, and a Visitor's Center where you can get a view of Mt. Whitney.
Lone Pine to Mammoth will take another two hours. There are plenty of accommodations there. Your teenagers may want to take the gondola from Mammoth Mountain Ski area to the top - 11,000. Some like the enjoy being at the lowest and near the highest peaks on the same day. Mammoth also has short hikes up at the lakes, and there is a Visitor's Center just as you enter the town for more information.
Mammoth to Lee Vining (Mono Lake) is another 45 minutes. You will enjoy starting out refreshed and having time to stop and enjoy the spectacular views along Tioga Pass.