Consult a local antique or rug restoration business. The owners will often have had a variety of training experience and can tell you about their educational background and the kind of work they complete in their business. Most often, people are trained in restoring antiques and furniture, so look into any antique stores in your area and ask questions. Ask them about where they were educated, how they came to work in their line of business and some of the techniques they utilize.
Research local schools that may offer courses in antique restoration. Many of these broader courses will train you in restoring everything from books to antique clocks, and textile restoration will often be a part of this broader training.
Research degree and graduate programs in textile restoration throughout the United States. Columbia University offers a broad program in art history and archaeology. The Smithsonian website offers a list of programs that train professionals and beginners in textile and antique restoration. These programs offer professional-level training, which can lead to a more focused pursuit of restoring damaged tapestries. These programs will offer the support to do research and practice on damaged tapestries, while also giving you the professional background to pursue a career in textile restoration.
Apply to programs that fit your goals regarding damaged tapestries. If you are more research-oriented, you might want to look for a school that specializes in writing. If you want to do hands-on work, then look for schools with emphasis in training.