Dave, you have a couple of options in the area. There are two "proper" trails up to Whiteface -- one is the Wilmington Trail (obviously coming out of Wilmington, NY) and the other is on the opposite side coming up from Lake Placid (the Lake not the town) -- called the Whiteface Trail from Whiteface Landing. The first trail can be shortened by starting up Marble Mountain, but these are on the far "other side" from where you will be.
There is a trail that comes up next to Lake Placid heading up to Whiteface Landing. The fun part of this approach is that you can paddle into the trailhead and go from there in the summer (maybe this winter...??). However, there is plenty else to do rather than bagging a peak if you can deal with that. One recommendation would be to take the Connery Pond Trail into the Pond or further past up the Truck Trail. This is just down the road from the Ski Center and the bus could certainly let you off on the way to the area. If you are done early, I'm sure you can hitch a ride down to the area. Most traffic is going there anyway. Also, I believe that there is a shuttle running out to the mountain, so you may be able to make arrangements or find out what they will do to accomodate your wishes. (The shuttle runs between the mtn and town.)
Connery Pond Trail (highly recommended snowshoe or X/C ski):
Head out toward Whiteface on Rt. 86. A couple of miles from town, you will see the trailhead on the left. If you cross a bridge, then you have gone 100 yards too far. Any driver should know where it is.
Look
HERE for a topo of the area.
For the record, the other "non-proper" options for Whiteface are the ski slopes mentioned by WalksWith above... or the slides
or hiking up the Whiteface Mountain Highway that comes up from the otherside of Wilmington.
The hike that Tony is suggesting is a classic, but you will be out in another direction from your group.