The best way to get cast on a Nickelodeon show is to audition for the show's casting directors, via a meeting set up with your agent. Without an agent, online resources such as Backstage.com are the best chance to find upcoming auditions.
There are a number of websites that release acting auditions for major networks like Nickelodeon. Backstage.com is the most well-known, but there is also ActorsAccess.com, which may also release auditions for Nickelodeon shows.
The most important thing to do is to research the network and get a sense of the casting directors it uses. Generally, the department or company posts the upcoming auditions. Then keep an eye out for information on upcoming casting releases, and follow the social media account to stay up to date on open calls.
Another way to get work as an extra would be to join a group such as Central Casting to get set experience, so you show up to your audition prepared, familiar with how to work on set and ready to win that part. Once you have your audition time, make sure to memorize your lines, show up on time, and be respectful of the casting directors and what they request.