The Butler Wash Ruins are only a 20 minute drive West of Blanding, UT and is only 1 mile RT, so it makes for a great "get-out-and-stretch-your-legs" hike.
7 Mile Canyon (sometimes spelled out, Seven) is located only 11 miles North of Moab, Utah. It's a great, short hike that dogs and kids will enjoy, as it passes by several petroglphys and pictographs. The trail is very easy and flat, so no hard effort is required to get to these artifacts. 7 Mile Canyon is on the way to Dead Horse State Park, but it definitely has a quiet and remote feel. During our hike we only passed one other person. This hike is very sandy, and you'll hike in and out of the dry steam bed. There's very little shade, so be prepared with the necessary items to keep cool, and carry at least 2 liters of water (and water for your pup!). Please keep in mind that touching the artifacts are not allowed.
Fish and Owl Canyons, located in the Cedar Mesa area of Bears Ears National Monument, is a popular loop hike for backpackers looking to discover ancient ruins and kivas, hike through desert terrain, visit Neville's Arch, and explore two scenic, deep canyons. This area is recommended for intermediate to advanced backpackers due to the distance, rugged terrain, light route finding and scrambling.
Teal Lake sits at an elevation of 10,400 ft, and is accessed via the popular Ruth Lake Trailhead. You will follow this trail for the first mile, and then after that there is no trail. You should be comfortable with a little route finding, and hiking off trail. The nice thing is that Teal Lake is a mere 2.2 miles, so you really only have to hike off trail for 1.2 miles. You'll pass a few unnamed ponds, Jewel & Naomi Lakes, then reach Teal Lake. There are several nice campsites at Teal Lake, and fishing is plentiful.
Elbow Fork in Millcreek Canyon is a popular trailhead for Mt. Aire, the Pipeline, and for accessing Lambs Canyon Pass, the dog-friendly way. Many people chose to hike up to Lamb's Canyon Pass via Lamb's Canyon, however that section of trail is part of the SLC Watershed, and dogs are not allowed.
Gilbert Lake is located in the High Uintas Wilderness on the East end of the range, with Gilbert Peak (13,442 ft) being one of Utah's "13ers" (a peak above 13,000 ft). It is the 3rd highest peak in the state, and the second highest county high point, and can be access by several drainages, with Henry's Fork being the most popular, however we accessed it from Gilbert Lake. The trail to Gilbert Lake follows the West Beaver Creek trail for 9.2 miles before reaching the basin which has four lakes total, great campsites, lots of wildlife, and of course, fairly "easy" access to the peak.
Mt. Hooker (12, 509ft) is located in the Southern end of the Wind River Range in Wyoming, and is home to a popular climbing route. The face of the wall is a sheer 1,800 ft granite face - it's one of the tallest and steepest vertical cliffs in Wyoming, and also 16 miles from the nearest road. Mt. Hooker is named after Joseph Dalton Hooker, a prominent British botanist and explorer. The granite face was first climbed in 1964, which took over 3 days to complete. The climb is rated at a 5.14a. Mt. Hooker towers over Baptiste Lake, a very deep and clear alpine lake which offers great camping sites and fishing.
Backpacking to Round, Sand, and Fish Lakes are part of the upper Weber River drainage on the Western end of the Uinta Mountains. The trail steeply follows the Dry Fork stream and canyon, as it makes it way to the three lakes. Camping and fishing are plentiful here. Round Lake is actually home to three species of fish (somewhat unheard of in the Uintas, as there are typically only two species in the lakes). There are Grayling, Brook, and Cutthroat Trout. Further ahead is Fish Lake, which can vary in depth by 19 feet, depending on the snow year and dam levels. Plenty of day hiking options are available too, as most day hikers try to reach the ridge for better views. This route is kid and dog friendly!