There is a spaghetti polonaise of trails on the hills to the south of Pokatello, generally know as the City Creek Trails...
Typically they are two-way shared tracks at intermediate level, about a foot wide, with a slightly concave smooth surface, undulating and with few technical features. The environment is generally wide open, with grasses in exposed areas and trees near the creeks and in and sheltered areas. The view over town to the north is expansive from the higher trails.
The busiest trail in the network is called Lower City Creek Trail and leads up the creek to the other trails. It's very busy in rush hours - perhaps a 100 riders between 5pm and 9pm on a summer evening - also runners and dog walkers. You can avoid most of it by swinging left onto the Bench Track about 300 yards up from the car park.
There are some signs to guide out-of-towners but you'll be left scratching you head at many forks. Not to worry, just explore as it's hard to get lost since you will usually have a view to the north.
For a more challenging ride head up to the advanced trail called "Over the Top". It's easy to build it into a loop which will give you about 1800 feet of altitude gain (and loss) over about 10 miles.
The City Creek Trails link to Gibson Jack, Slate Mountain and to dozens of other trails in the hills further south. All up, the greater network adds up to more than 50 miles of varied singletrack.