Take my photo, take a photo of me

Both are possible, depending on context. In BrEng both 'picture' and 'photo' occur (but you don't often hear 'snap' any more.)

"Take his photo" implies to me that it could be for a specific purpose or part of a process (for a journalistic reason, for documenting something, etc.):

The police took his photo, then took his fingerprints.

Last week at school, they took my photo for the yearbook.

"Take a photo of him" is used when it's done more in the casual, spontaneous sense of photography:

We took a photo of the protesters during the demonstration.