You can just say ¿Cómo te llamas? or ¿Cuál es tu nombre? to ask someone name. If you want to ask the same in a more formal way you will say ¿Cómo se llama? or ¿Cuál es su nombre?
Usually, in an informal conversation, we will reply just with our name «Me llamo Lola», while in a formal situation we will say our name and surname «Me llamo Lola Sánchez».
Notice also you don`t need to repeat the verb «llamarse» in your answer, you can just say «Lola» or «Lola Sánchez».
Also, you can ask someone directly about their surname ¿Cuál es tu apellido?
Finally, remember that it is going to be very important to learn how to spell your name (practice the alphabet in this link) since most people are going to ask you how is written or spelled«¿Cómo se escribe?» or «¿Cómo se deletrea?»