you can ride from Taylor Park to within 7 miles south of Crested Butte. All the roads into Crested Butte itself are posted No ATV's. So you can't connect to any of the other Crested Butte trails from here or any other point that I'm aware of.
From Taylor Park ride the Italian Creek Road (759) to the Reno Divide Road (759) to the Cement Creek Rd (740). Just before the Cement Creek Rd hits Hwy 135, at the NFS boundary, no ATV's beyond this point. Or from Taylor River Rd you can use FS 748/744 Reno Ridge Rd to get to the same point above at Reno Divide and 759 down to 740.
The only trails through the forest connecting Cement Creek to Brush Creek Rd, which is a little closer to town, are motorcycle only designated trails.