Pero and sino can both be translated in English as” but”.  However, they have slightly different meanings in Spanish.  

PERO

Normally translated as “but” in English

Pero usually joins the two parts of a sentence where each part or phrase could stand alone.

If the sentence starts with a positive then pero will be used. 

If the sentence starts with a negative then pero will not be used.

EXAMPLES

Tengo que irme al colegio pero no quiero.  I have to go to school but I don’t want to.

Quiero ir al colegio pero no puedo.  I want to go to school but I can’t.

SINO

Normally translated as “rather” or “ however” in English

Sino usually indicates that one part of the sentence is negative to the other.

If the sentence starts with a negative then sino will be used.

If the sentence starts with a positive then sino will not be used. 

EXAMPLES

No quiero ir al colegio sino quiero ir al cine.  I don’t want to go to school however I want to go to the cinema.

No me gustan las hamburguesas sino me gustan las papas fritas.  I don’t like hamburgers however I do like chips.