<map lat='41.7259' lng='-111.834326' zoom='10' view='3' float='right' /> Logan is the largest city in Cache County, Utah. It's said to be the "Safest Metropolitan Area" in the United States. The Cache County area is located between Yellowstone National Park and Salt Lake City, which makes it a great pit stop for road trippers going up and down Interstate-15. The city has free public transport throughout much of the area, so it's very easy to get around. Although it's said that hitchhiking is illegal in Utah, to police generally have better things to do than harass hitchhikers, unless they are blatantly doing something illegal (standing on the interstate, etc.).
South towards Salt Lake City, Brigham City
Highway 89 heads south from Logan to Brigham City, passing through "Sardine Canyon." There are two options for hitching, and either work great.
Option 1 Take Bus #7 to the McD's (it's last stop before it turns around), and walk south along Hwy 89 about 3 to 5 minutes to pass the businesses. Thumb from there with or without a sign. 
Option 2 Another option would be to take Bus #11 to Nibley and hitch from there. However, it's really faster to use bus #7 and hitch by the golf course.
North towards Idaho
If you're going to Idaho, it's best to use the "Cache Valley North" (CVN) bus that goes 13 miles north to Richmond, Utah. Take that, get off at the last stop and walk about 10 minutes north on Highway 89/91 to get out of the town. Should be easy from there. Watch for cops however. 
West towards Tremonton
Take bus number 8 to 1000 West and 200 North. There is a gas station there where you can ask for rides from people filling up their tanks. A sign and neat appearance works well here. You can also try thumbing directly on 200 West, but it's copville since the county prison is nearby. 
East towards Bear Lake, Garden City
Take bus #1 and get off right before it comes back to the university the second time (so before it turns right on 700 North). Walk East, down Highway 89 (you will walk past the golf course). You can hitch from the road, but cars are usually going too fast, so it's best just to walk all the way down to "First Dam," and hitch directly on the bridge. Besides, it's beautiful scenery, it would be a pity to not walk all the way down. 
South towards Paradise, Huntsville
Take the "Cache Valley South" (CVS) bus to the corner of Hwy 195 and Main street in Hyrum (Mountain Crest High School). Walk south along the highway passing the school, then start hitching right after the corner, before the street becomes narrow. 
The Cache Valley Transit District buses run every half hour normally, but some lines run every 15 minutes to and from Utah State University. It's free to ride the buses; keep in mind that they stop running at 10:30pm.