One of the best features of HotS is the ability to play unranked. The reason I don't like playing multiple races on ladder is because it can screw up my MMR, causing me to not know if I'm actually figuring out the race or just playing weaker opponents. This should help greatly if that is your issue as well.
I would recommend focusing on one race at first. Playing random is fine, but I think you have to focus on specific stuff to really learn, so even playing one style as a race is important before you branch out. Once you've explored all these things I think random can definitely be an option, but I wouldn't recommend doing it at first as it is hard to make iterative improvements from match to match (seeing the same matchup can take many games and you might forget what you wanted to try out / change.)
EDIT: PS) This is the right forum to post, good job. :)