It is important to encourage your child to explore their gift. If your son or daughter is an artist, try to keep supplies around the house. If you child is a mathematician, do your best to find them challenging new problems to solve.
Unfortunately, there's a fine line between encouraging your child and being a bit of a stage mom (or dad). Don't push your child too much, so as to create a negative environment for the young person or, in worst case scenarios, jeopardize their desire to continue learning. Try to support them without continually putting them on the public stage, and let them determine how they spend their free time.
Schooling can be an issue for gifted children as well. In the beginning, let the school do its job. Most times, teachers can assign gifted students special projects to keep them challenged. However, if there's a problem and your child is bored, investigate the issue. Sometimes bored students will display behavioral issues that can turn into problems later in life.
If what the child is being asked to do in class is too easy, there are a few options. You can ask the teacher to give the child independent work at their own level. Or you can ask if there's a more advanced class at the same time that he or she can attend. Grade skipping is another option, but it's more complicated, because a child may be gifted in one specific area and more average in others.
Unfortunately, there are very few schools that are specifically for gifted students, and sometimes parents still complain that their children aren't challenged. Though it may be tough when your child is in elementary school, as your child ages, more options open up at public schools, and there are advanced placement classes and international baccalaureate programs. Competitive prep schools often target gifted students, so these are also an option. Once kids get to college, it will be easier for them to find their niche.
If you're concerned about your child being challenged in school, there are things you can do outside the classroom, too. Be your child's advocate, and if the school is not willing to make accommodations, do everything you can outside of school -- science museum courses or classes at the local art institute -- to pique their curiosity.