Utah State University is a very large school, and its students probably rent more apartments in Logan than any other one demographic. As such, you have to respect their schedule when apartment shopping. Realize that most students probably leave town around April and May, and that this is the best time for you to do your own apartment shopping.
When you're applying for an apartment in Logan, remember that this is a typical Utah city, which means that religion is very important to many people. Not everyone is religious, and not everyone is Mormon, but there will be a smattering of landlords out there to whom religious affiliation is very important. It is best not to mention the topic just to avoid accidentally offending someone.
Getting around Logan is very simple. There are free busses that can take you through Logan, or to some of the surrounding communities. Many people bike or walk all throughout the town. One location that is difficult to reach without a car or the bus is the University. It is at a significantly higher elevation than the rest of the town, and the trek can be arduous without a car.
Like much of Utah, the countryside in and around Logan is spectacular. Willow Park is a well-groomed park with a swimming pool, and if you're looking for something a bit more rugged you can head to Logan Canyon. In the winter, you'll also find the local ski hill on the north end of the Canyon. You can also find almost an endless variety of challenging hikes in the surrounding area. The White Owl is a great place to grab a beer, especially on the upstairs patio.