Go to Broadway Shows. Every Broadway season, even A-list movie stars can be found on Broadway in high-profile shows (Al Pacino recently did "Merchant of Venice"). Most theaters will have a stage door that fans will line up outside of to catch stars on their way in or out of the theater. Though it's not guaranteed that the stars will stop to interact with fans in any way, this is a very consistent way to at least encounter celebrities.
Research book tours. Celebrities often write books, which are accompanied by national and international book tours. At each stop of these tours, fans can often get signed book copies, photos with celebrities, and the chance to chat one-on-one with the author.
Go to conventions. Giant film and comic conventions such as Comic-Con are packed with the stars of popular superhero, comic book and action films and TV series. Furthermore, these stars go to these conventions expressly to interact with fans and promote their work, so there are many question-and-answer panels, autograph signings, and photo sessions. These large events usually cost money and require advance registration to attend.
Visit movie sets. Movies are constantly being filmed all over America and the world. This is not the most consistent way to meet stars if you live in out-of-the-way rural places, but you'd be surprised where films set up and shoot. Films often incorporate landmarks and businesses in small local communities for establishing shots and scenery. If a movie is filming near you, it may be worth your time to visit the shooting locations, given the possibility of running into one of the stars.
Take studio tours. Most major production companies and film studios offer tours of their facilities. There are fewer opportunities to meet stars via this method, since the tours are usually designed not to interfere with active production and artistic work, but you stand a much better chance in this setting than you do if you're hoping for a chance meeting on the street.