For general discussions among believers, Satan, Lucifer and the Devil are synonymous. However distinctions are sometimes important.
Satan does not appear in the Bible until the time of 1 Chronicles or Job, long after Moses reportedly wrote the Torah, and also after the period of the Divided Kingdoms. In Job, Satan is not a proper name but a title: "ha-satan," the adversary. This becomes important for understanding, for example, what Jesus meant when he said "Get behind me Satan," (Matthew 16:23) to Peter. We have it in Greek, not Jesus' own language, and even in Greek there were no capital letters in the original. So Jesus may well have scolding Peter for acting as adversary, but the way it reads in English he could easily be saying that Peter was literally inspired or possessed by the Devil.
The name Lucifer is even more illusive that Satan in terms of its evolution. In Isaiah, it does not refer to Satan or the Devil. It refers to the actual King of Babylon.
You will take up this taunt against the king of Babylon:
The Lord has broken the staff of the wicked,
the scepter of rulers, that smote the peoples in wrath
with unceasing blows, that ruled the nations in anger
with unrelenting persecution... Your pomp is brought down to Sheol,
the sound of your harps; maggots are the bed beneath you,
and worms are your covering. How you are fallen from heaven,
O Day Star, son of Dawn! How you are cut down to the ground,
you who laid the nations low! (Is. 14:4-12)
Christians would later interpret this to refer to the Archangel Lucifer, a name closely related to the Day Star in the original. But the proper name Lucifer appears nowhere in the Bible.
The Devil is another matter. The word does not appear in the Hebrew OT, which speaks of demons. It comes from the Greek "diabolos" which can translate as "accuser," and thus shares a meaning with "ha-satan." Devils were also spoken of in Jewish literature of the pre-Christian century in Greek, including several arch-demons, though not Lucifer. There are dozens of examples of its use in the NT. Many of these make it clear that the Devil is both the "adversary" and "Satan.
In everyday use, the three terms are definitely synonymous. But it is useful to know the background of the terms. For example in speaking to Jews, they will certainly know what a Christian means when when he talks about Lucifer but the story of Satan/Lucifer being cast down and conquered by St. Michael (Rev. 12:9) is a Christian tradition, not a Jewish one.