I also recommend Farragut State Park. It has well-designed and well-maintained campsites in a forested setting and is located close to the area's major attractions. However, if you venture north to Sandpoint, beware of the view of the town while crossing the long bridge over the Pend Oreille river. It has seduced many into pulling up stakes and relocating to this picturesque resort community by the lake.
You also might consider the Alpine Country Store and RV Park just north of Coeur d'Alene. (http://nirvpark.com/) It does not have fancy amenities, but it has numerous shade trees with larger than average green spaces between the sites.