Different senses and sensory integration problems show up in different ways. To fully learn more about each sense and the behaviors that indicate a sensory issue, read The Out of Sync Child. This author also has a book for teachers that is composed of checklists and examples of strategies to use in the classroom. It's a great gift to give your child's teacher!!
One red flag I look for to indicate possible sensory integration issues: a child who has meltdowns frequently and has a difficult time getting it back together or being consoled, or whose meltdowns seem really intense and extreme for the incident; a child who is bouncing around the room, throwing everything hard, jumping off of everything, running into things, knocking blocks over, making lots of noise with the toys, moving frequently- these are signs of a child craving deep pressure activities.
Deep pressure activities include: carrying a backpack of books or heavy items around, pushing a laundry basket down the hall or carrying it, pushing or pulling a wagon loaded with heavy items (rocks, a bag of soil, bags of garbage or leaves), moving books in the room, putting chairs on desks, carrying the garbage out, pulling the garbage cans down the driveway, clapping erasers, erasing the board, walking up and down steps with a backpack on your back, push-ups against the wall, jumping jacks, jumping on a trampoline ( or a small exercise one) bouncing on a hippity hop, bouncing and throwing a ball.
Provide your child with the sensory "diet" they need- look for the activities that give your child the sensations and information their body iis seeking.