Apartments in Rexburg are in short supply, so you need to try and look when there are the most places on the open market. Usually, this is around the mid-summer mark. July is a very good time to shop, but June and August are both pretty good options as well.
You should take the conservative reputation of Rexburg into account when meeting with prospective landlords. Just like you wouldn't move to San Francisco and expect to make friends by announcing that you're a staunch conservative, you could easily alienate people in Rexburg by taking a harsh liberal stance. Just keep the conversation polite and on topic and you'll avoid accidentally stepping on any toes.
Driving is certainly the preferred mode of local transportation. Many people also simply walk, as the city is compact enough that you can walk between many major locations. Traffic volume is low and parking is usually readily available though, making Rexburg a very car friendly city all around.
The strong influence of the church in Rexburg means that there are very few bars, and essentially no nightlife. It is only recently that you could start to purchase drinks in some restaurants. Outdoor entertainment in the area is great however, with natural features like the Goldbug Hot Springs and the Darby Wind Caves making for a great outing. There is also a popular series of demolition derbies held during the spring.