I was on the depo shot for approx. 8 years. =)
You are not 100% protected until the end of your first shot cycle, which is 90 days. During the first 3 months, you'll probably do a lot of spotting and weird stuff, don't be alarmed - it's normal.
And no, you don't have to be on your period to get the shot or to start it. It's prescribed for a lot of women who have major issues with irregular bleeding, because it usually stops your period altogether. I bled about the first 3 months, on & off, and after that I didn't have a single period, spot, or anything for the remainder of my time on the shot. =)
So long story, yes, it'll still be effective. HOWEVER, until you get that 2nd shot, make sure you use a backup method!