I know that Peter is the first Pope, but I do believe that he is called a bishop and is one of several including the Bishops of Constantinople, Jerusalem, Alexandria, and others. Who was the first Pope to call himself by that name? Was it Leo who increased Papal power or someone else?
The first title to be used for "pope" properly was the Latin Pontifex Maximus, which was originally the title for the chief of the College of Priests (Collegium Pontificum), sort of religious priests of the city of Rome itself.
By the time Christianity came around, the Pontifex Maximus of Rome was the emperor. That changed with Gratian who renounced it. Later, Leo I (as you guessed) resumed the title.
Papa (Gk. πάππας, "pappas") from which comes the English pope, was applied to senior priests, but seems to have been applied to the chief of Christian bishops only with Pope Heraclas, if Wikipedia is to be believed.
There is a disagreement between different authorities. Some consider Clement to be the first legitimate successor of Peter the Apostle and the first Bishop of Rome after him. The Catholic Church itself considers Clement to be the first pope after Peter. The reason for this is that the Liber Pontificalis, the official book on the popes of the Church, specifically says that Peter chose Clement to be his successor. Other authorities consider Linus to have been the first pope after Peter..