By Interstate, I'm guessing you're looking at I-70? You definitely do not want to be on I-70 there, and it's probably illegal, but it's easy to avoid - it's even a nice ride!
The area has a fantastic series of bike trails. You don't really need a map, but I bet you could find one online somewhere.
From Silverthorne, you can get on Dillon Dam Road and ride west along the north shore of the reservoir. You may be able to get on a bike path around there, but if not the road is OK. When you get to Frisco, go south into town (rte 9 is ok) and go to the southwest edge of town (ok to take Main street, or just wander thru town, I recommend the pastry at The Butterhorn on Main and the beer at the brewery on the corner of Main & Hwy 9 - I watched TDF there a few years ago). (I digress.)
The bike path to Copper Mountain skirts the edge of town on the southwest side. Just find the bike path (really just ride to the southwest edge of town, you'll find it), get on it heading west. It parallels I-70 (it actually shows up as a road on google maps, but it's a bike path) and dumps you out across the street from Copper Mountain. You'll ride right by a gas station with a Quiznos and Starbucks.
Highway 91 from Copper Mountain to Leadville is not an interstate, and is fine to ride. The first part heading up from Copper doesn't have much of a shoulder, but it improves.
Leadville has lovely camping at Turquoise Lake, a really friendly hostel, and full services.
(edit, clean up some cut-and-paste error)