There are a lot of hiking options: 

In the Timber Lakes cabin there is a huge wall map of the local area that has highlighted areas showing where the local trails are- and there are some really cool ones; that said let's set an expectation- these hikes are under a mile and are all about the nature, not a destination.

 

If you want forested single track style hiking there are trails in Dutch Hollow in Midway, Deer Valley, Sundance, Provo Canyon and Park City, as well as a few hundred options in Salt Lake. 

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There are dirt roads in the National Forest that also access many areas of fields, meadows and the like that become blanketed in flowers- summer can really be stunning in the backcountry. The National Forest by the cabin is not known for it's hiking trails, it is more of an "outdoorsman" area with dirt roads, forest and lakes.  Lots of hunting, lots of camper camping up there, not the backpacker/ hiker style trails that Salt Lake is known for.