I used to play random and in order to improve, I have also forced myself to stick to one race. At first I picked zerg, but then I realized that I strongly dislike the ZvZ match up. I also didn't like TvT. Of the three mirror matches, to me PvP is the least aggravating, so I decided to play Protoss.
Perhaps choosing the race with which to maximise your enjoyment in all three constellations of opponents could be an idea.
All the best,