Assuming that the anime follows the manga's story line pretty closely, I will usually tend to catch up with a series by following the anime until it either enRAB or I catch up to the point in which the currently running episodes are at, in which case I will then start following the manga from the point at which the anime last left off at in covering the manga's story line. I do this because if they have the same story, I might as well see it in animated form if I have the oppurtunity, since I won't feel like bothering to check out the anime if I have read the manga and therefore already know everything that will happen (aside from what happens in potential filler material that may be used in the anime).
A good example for a series that I've done this with is One Piece. I'm still following the anime episodes for the series, but I only started reading the manga from the point at which I caught up with the current anime episodes, which just happened to be perfectly at the end of the Thriller Bark arc (it was currently airing fillers at the time, and had not yet started on the next story arc from the manga). I picked up reading the manga from that point, and read all of the way up to the beginning of the Impel Down arc, which is where the manga was currently at, during the point in time in which I started reading it.
Now, if the stories differ between the manga and the anime by a significant amount, then I usually tend to read the manga first, and then see how the anime's loose adaptation of that original story line turns out.
A good example of this is Fullmetal Alchemist, and I'm referring to first reading/catching up to the manga and then watching the original TV series (since this was long before Brotherhood came out); granted, that I was technically first introduced to the series by watching a few random episodes of the anime when they first aired on Adult Swim, but I don't really count that as watching the anime before reading the manga, since those episodes took place in the middle of the series, and I didn't even really understand what the story was about, at the time that I watched them.