* ¿Hay escuela hoy? This translates literally to "Is there school today?" and is a common and natural way to ask.
* ¿Tengo clases hoy? This translates to "Do I have classes today?" This is a more personal question, focusing on your own schedule.
Both are correct, but the best choice depends on the context. If you're asking generally about whether school is in session, use "¿Hay escuela hoy?". If you're asking about your own classes, use "¿Tengo clases hoy?".