You will need a City of Surprise business license. You would be wise to also have some corporate formation documents. Forming an LLC would provide you with a way to limit your personal liability.
There are typically two types of licenses: a business license from the city and a professional license from the state. Whether or not you need a business license depends on the city in which you conduct your business, not the city where you reside. You do not need a license in order to create a website. One issue is to whom you would be teaching sign language: if it is to folks with a disability this might be deemed "occupational therapy" and might be regulated under Chapter 34 of Title 32 of the Arizona Revised Statutes (i.e., A.R.S. 32-3401 et seq). If that is the case you may need to be licensed as a therapist under the Chapter by the State. If, however, you are teaching sign language to non-impaired individuals, then you likely do not need to be licensed under the Chapter.
Nothing in this response is intended as legal advice to you, may not be relied upon by you as legal advice, and does not establish an attorney-client relationship.