There are many areas to hike from well-laid out and marked trails to very difficult unmarked ones. Popular ones with families are Shawka, Fossil Rock, al Rabi, Seven Summits, Abadila, Tayyibah, Tawyeen and Jebel Jais. You can find information on all these trails and more on Wikiloc.
Please do your research before going and ensure all safety precautions are adhered to as there are rescues from mountains nearly every weekend of inexperienced and/or ill-prepared hikers. There are many groups that offer organised hikes with experienced guides.
Wadi Abadila, Fujairah
Jebel Jais Upper Segment
Jebel Jais Lower Trails See more here.
Seven Summits, Fujairah Take me to the map
Al Rabi, Khor Fakkan Take me to the map
Tawyeen, Fujairah (upper part)
Ain al Sheria (beginning of trail), Fujairah Take me to the map
Bidya Fujairah Adventures See more here
Masfoot Trails Take me to the map