How to create new taxonomy and add to search form filter on the main page?
Easiest way is to create an attribute and then make it searchable on search bar. In the edit attribute check mark this:
Display in the search form
Yes, you can create a listing attribute, e.g., select and mark it searchable, then taxonomy will automatically register, and you can add options to this attribute and search by them without additional code. Please check this doc How to add listing attributes - HivePress Help Center