Что мне делать сегодня?

<<Что мне делать сегодня?>>

Does this mean, "What do/should I do today?"
Is it using the dative case?

I'm lost. :|

Yes, мне is in the dative case. Russians often use these infinitive sentences for "should" questions. The subject of the infinitive then shows up in the dative, although it is often left out as well. Examples:

Окно открыть?
Should I open the window?

Что купить на рынке?
What should I buy at the market?