When you say, 'Is he mostly Phoenician? Turkish? Greek? African?', I have to emphasize that the utility of a DNA test rests to a large extent on the distinctiveness of the ethnic groups you are comparing. If I 'listen' carefully to what you are saying it appears that your father has some Middle Eastern or AfroAsian characteristics that you would like to know more about. You could do an autosomal DNA test that will give you an approximate percentage of his various ancestral admixtures. This then needs to be interpreted and may not be very conclusive as Lebanon, Turkey, Cyprus and Egypt share a great deal of DNA and your father's paternal ancestry may have experienced a fare amount of introgression from having lived on Cyprus for many generations. This introgression means that any foreign DNA would become negligible over time as every generation crossover occurs between the foreign markers and local markers.
FTDNA currently has a sale of $90 on their Family Finder test and they post kits anywhere in the world for 5 dollars.
More specifically, if your father looked European then you could test your yDNA haplogroup as most European males fall into a few distinct groups. If I was you I would look at genealogy first, then family history and finally bring in DNA testing.
Good luck!