If the teaching position requires a bachelor's degree and you have that degree, the H-1B is the best route. You should definitely have a lawyer to help with this. The employer does the filing, not the employee. You couldn't start working until October 1, unless the school is a non-profit affiliated with an institute of higher education. The costs vary, depending on the type of school because government filing fees vary with the size of the employer. It is likely to cost a total of $3000 - $5000.