Like Singapore and Hong Kong you can also travel Dubai via Dubai Metro Train from the airport. Dubai International Airport Terminal 1 and Terminal 3 are directly connected to the metro station.
Don't worry about the luggage, there are spaces designed for suitcases and big baggage.
For a 20 AED Nol Card partially loaded of around 15 AED you can already take the train and see the beauty of Dubai from the panoramic windows of the train. Almost every shopping malls are also connected to every train station.
Every train station has a feeder bus, and a taxi that will bring you to your destination.
Burj Khalifa, Burj Al Arab are sure to see when you are in the train.