Edna. One of the articles of Iman (Islamic faith) is to believe in all Prophets which includes Musa (Mosses) and Issa (Jesus),peace be upon them. Not only to have belief that they are Prophets but also to revere them all. Islamic faith also constitutes another article which is to believe in Holy Books - meaning thereby believing in what(Book) Mosses brought and that which Jesus brought. Please note that ISLAM DOES NOT TELL THE MUSLIMS TO FOLLOW MOSSES AND JESUS, instead, Islamic-teaching is to follow THE LAST OF THE PROPHETS, MUHAMMAD (PEACE BE UPON HIM) AND TO FOLLOW THE BOOK HE BROUGHT, THAT IS QURAN. I would also like you to know that Muhammad (peace be upon him) is the last of the prophets till the end of the world, meaning thereby, that Islam is the final Deen [code of life as prescribed by Allah(God)] for the whole of mankind till the Doomsday.
About the last part of your question, Jews have Saturdays, christian have Sundays and Muslims have Fridays as the holy days. I would like to tell you that the All-Mighty God had chosen Fridays as the holy days for all the Scripture holders, Jews changed it to Saturday and christian to Sunday. This example alone could stir up one's mind to think what other changes might have been brought about in the previous Commandments of God, by the respective Scripture holders with the passage of time, that necessitated the renewal of God's Commandments, by God, through Prophets after Prophets till the last.
You want to know more please feel free to ask.