It depends on the story, but for the most part the manga and anime are usually differant enough that I will usually try both if I like the story (as with Planetes for example).
As for spoilers - well you'll get them regardless for the other one no matter which you start with.
I'm more inclined to go with whichever medium I have the whole thing of first, then go over to the other one, rather than trying to follow both simultaneously.