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Acting is a specialized type of performing arts, and it generally requires a great amount of skill. Although many actors are naturally talented, it often takes some type of formal training to get the most out of these talents. Because of this, many aspiring actors choose to attend performing arts or acting schools.
You can also check the below links to check for some top universities and have a comparison on which one would best suit you.
Also here are some drama schools attended by well known actors .
Scott David’s Answer
NYU Tisch school of the Arts, Carnegie Mellon, Juilliard, Yale Drama, and Pace University.
Those are on the top of my list for straight acting. Musical theater is different. However, in New York there are also conservatories for acting, such as Lee Strasberg Institute, William Esper studios, Atlantic School of Acting. It is possible to attend NYU and partner with one of these acting studios as well.
I would recommend going through a university to attend a conservatory, that way you are not paying out of pocket, but financing a student loan. Also New York is expensive, and not the easiest place to live. Be sure you are fully committed before making the jump, because study is a long time away from home. However, if you love the work, you will find a way.