You've asked an interesting and useful question. I don't have a definitive answer, as I think both views are valid.
To start, Pirkei Avot (Chapters of the Fathers) Chapter 5, Mishnah 21 cites Yehudah ben Tamai who said that when one turns 10, he should learn Mishnah, and at 15, he should learn Gemarah. Perhaps, this is the source of separating the two. I hunted, but I haven't found a commentary that elaborates on why Yehuda ben Tamai said this. But, apparently, the Mishnah, encourages, or at least Yehuda ben Tabai, himself, suggests, separation. I surmise that since he places ages, his opinion is based on mental maturity / capacity.
So, what I state below is my opinion, based on having done both methods - together and separately.
The majority of the Mishnah is in Hebrew and most of the Gemarah is in Aramaic. In elementary school, I learned Hebrew first, so in 5th grade, I studied only Mishnah. In 6th grade, prior to learning Gemarah, we had to memorize several pages of common Aramaic Gemarah words before we could even begin learning Gemarah. Bear in mind, that this was in the pre Art Scroll era. (I'll address this, briefly, later.) In hindsight, while my rebbe was harsh, and a bit meshugah in his pedagogy, I think this method did get us to better understand the gemarrah. So, in short, if your Hebrew is good, but your Aramaic is next to zero, you MIGHT want to follow some of my original methodology.
Having stated this, we are now in a wonderful era of Art Scroll, Soncino and Steinzaltz. I prefer Steinzaltz, even in the Hebrew version, though the English one is wonderful for beginners as well. Point is, that even if your have no Hebrew or Aramaic knowledge whatsoever, and even if you can't read Hebrew, you can understand Gemarrah well.
Preference? I find that learning Gemarah together with the Mishnah is better, esp. if you have a good command of Hebrew and enough of a command of "common Gemarah Aramaic" to follow the most common words as well as understand Gemarah methodology (it's rules). The Mishnah is an "overview". It states the facts along with different opinions, sometimes, of what to do. Rarely does the Mishnah state the Torah source or describe the details of how or why it said what it did. As a matter of fact, often when you look at Rash"i commentary on the Mishnah, he will say nothing more than "The Gemarra will explain this, further." So, in a sense, you're FORCED to view the Gemarrah! And even if you used Steinzaltz, Art Scroll, etc., for just the Mishnah, where do you think most of their explanations came from? The Gemarah, of course!
(I am, generally, not in favor of someone using any of these English translations, as, no matter who good they are, anything translated can not be as good as the original source. Something is almost always lost in translation. However, I do understand that not everyone is competent in English, Aramaic or "Yeshivish", so these translations certainly are useful for this crowd. But if possible, better to learn Hebrew and some Aramaic. Don't learn "yeshivish", though ... PLEASE don't. I get irked when people call them "tallaysim" and people stay "BY" someone for 3 "Shabbosim".)
Summarizing - If you're satisfied with a quick overview, the Mishnah will suffice. But, if you really want to appreciate what you are learning, understanding better the source of things as well as seeing different opinions and gaining a better appreciation of logic and honming your logical skills (a side-effect o following Gemarah methodology), then learn it with the Gemarah.
One last small note - There are many tractates that have no Gemarah! In the order of Zera'im, only Brachot has a Gemarah. As a matter of fact, the phrase I quoted at the top, from Pirkei Avot, is another tractate with no Gemarrah. So, if you were to just concentrate on Gemarrah ... my oh my... you'd be missing out on quite a bit :-) Just something to keep in mind!
Note: Yerushalmi includes many of the tractates left out of Bavli...