Hi Tulip,
The Madrid Metro system is extremely efficient, clean, safe -- not a big pain to use at all. It is cheap and you can get a 10 trip ticket to save time -- taxis get very expensive. The Palacio Real is near the stop Opera, the Prado is at Banco de España. Those are both on the red line and only a few stops apart. I have no idea where your hotel is, but Puerta del Sol is also located on the red line and that is pretty much the center of Madrid life, so if your hotel is centrally located it might be somewhere near one of these red line stops.
As for safety just be aware -- hang on to your bags and dont leave a backpack on your back with important stuff in it-- put it on your lap or over your front instead. The same sort of common sense stuff for any city. The only time I would not recommend using the Metro is if you have big suitcases with you because inside the tunnels there are lots of stairs, it is absolutely awful to lug lots of possessions up and down and then run to the train.