Homeschooling can be a challenge even for a college trained professor. However, it is very rewarding and children can learn at their own pace and learn anywhere in the world. The first thing to determine is where is your child in education now. You can have your child take placement tests to determine the level of education that they need or you could just continue where the public school left off. Once you decide the level of education you have to decide are you going to develop the curriculum or are you going to use a online school? To answer this you must decide if you have the experience and the patience to develop a curriculum. It is easier to develop for the younger child but as they get older it becomes more difficult. The high school student will need to complete certain requirements if they plan on attending college. The solution I recommend is to use an accredited online school. This gives you a curriculum, lesson plans, books and a school for your child to participate in. The top online schools use certified teachers for course work. Your child will participate in class with a teacher and other students much like brick and mortar schools but they could be anywhere in the world.
As far as socialization for the teen that is easy. There are local home school organizations that your teen can participate in which have proms, sports, clubs and other activities with kids their own age. So don't let anyone tell you that your child will grow up as a misfit because they were home schooled. The other thing to consider about homeschooling your child is the fact that they will not be around the constant influence of other people. They won't have to learn about the modified history of the planet, or revised economic plan but they can learn about anything that you chose.