Tantrums already?

Could we possibly be dealing with tantrums already? Lilliana is 15 month's going on 2 and I think is already showing quite the 'tude at times. Of course, this may seem astonishing to many of you because thankfully she is quite the little angel when we are out in public but when it is just Mommy and Lilly, she is in full swing. She has started throwing herself on the floor lately and that's really about as far as it goes. There isn't any leg kicking or flailing around associated with it but she is typically crying. This seems to happen when I take something from her that she is not supposed to play with or if I walk away from her too quickly and she wants to hold my hand or be carried. To be honest, I am not really sure how to handle it. So far, I have just been kinda ignoring her and sometimes laughing when I just can't hold it back any longer. I know that seems mean but she is just so dramatic at the ripe old age of 15 months. I have no idea how I am going to handle the terrible 2's and I have heard 3 is worse than 2! Any suggestions would be greatly appreciated. I just don't know if I should punish the acting out behavior or ignore it and not bring attention to it.