gif is fine as long as you set the background as transparant. it wont assume that you dont want the white.
.png is good for transparancy too. pictures are of a higher quality but equally of a higher file size
Nothing personal, but I question your ability to design logos if you're asking this sort of question...
And no, it's not enough to just save it as a gif. Save the logo as an ai file (i.e., a vector-based image format, not raster), and just design the logo with no background. Even though the background on screen is white, it should be transparent (it is by default anyway).