Learn Polish: Czy wasz pociąg jedzie dzisiaj?

Wasz is the plural "your" (like "you guys'" or "y'all's"). You use it when you are talking to more than one person. If you were asking just one person about their train, you would use twój. It takes the masculine form wasz (instead of the feminine wasza or neuter wasze) to match the masculine noun pociąg (train).

Why do we use jedzie for "going" here?

In Polish, you must choose your verb based on how something is moving. Jedzie comes from the verb jechać, which means "to go by vehicle" or "to ride". Since a train is a vehicle, we must use jedzie. If a person were walking, you would use a completely different verb: idzie (is going on foot).

Could I drop the word Czy and just make it a question with my tone of voice?

Yes, in casual, everyday speech Poles often drop Czy and just say Wasz pociąg jedzie dzisiaj? with a rising pitch at the end. However, starting with Czy is the standard, grammatically complete way to ask any "yes or no" question, so it is the best way to practice right now.

I sometimes see dziś for "today". Can I use that instead of dzisiaj in this sentence?

Absolutely! Dziś and dzisiaj mean the exact same thing and can be used interchangeably in almost any sentence. Dziś is simply the shorter version of the word.