Hey, PM me, my bike may or may not be back in service by then, but we can at least grab some coffee or something while you're here and do some ride planning!
Hit up the Mid Mountain Trail - it's also called the 8000' trail since it kind of hugs the 8000' contour line. Not techy, just smooth and some nice cruiser sections. You can access it from Deer Valley Silver Lake Lodge (free bus!) and ride to the Canyons, take the Gondola down or ride down, and then bus back for a fun 3+ hour loop.
From the Park City base, the Sweeney Switchbacks are technical, but can be a lot of fun. Same with John's Trail - it's a bit tight but again very cool. Most of the trails there are solid, and vary from open ski run to tree-enclosed. Depending on the group, Lost Prospector is another easy cruiser in the area that's got some flow to it.
Avoid the Rail Trail, it's a paved then gravel trail on an old rail bed, really meant for jogging and cruiser bikes.
Riding the lift-served trails at Deer Valley can also be fun, watch out for the Black trails though as they are hard to even walk down, let alone ride on a cross-country bike!
You really can't go wrong...drive out to the Glenwild area and ride Bob's Basin, Flying Dog Loop, Cobblestone. Those are very flowy, not much in terms of technical. More of a grass / scrub brush environment. Try Round Valley for some fun loops.
Some maps below:
http://mountaintrails.org/parkcity-trails/
