Tanks are the rarest role, followed by healers, then DPS. That will vary some from FC to FC, but in the overall game population, that's how it stands. Early on, healers are a little rarer; CNJ is the only class allowed to queue as healer before SCH becomes available at level 30, so the ratio is a little skewed.

Something to be aware of (if you aren't already) is that you will have to group for dungeons/primals at various points as part of the main story quest. The first set comes around level 16, 17, 18, with the first three dungeons, then the story mode Ifrit fight at 20. You can use the Duty Finder to get the groups, or you can shout and/or ask for help from your FC, but you can't solo them. Assuming no big problems, most dungeons take 30-45 minutes once you get in. If you're not a tank, you can tack on anywhere from 10 minutes (low level healer) to more than an hour (mid level DPS) for finding a group if you're using the Duty Finder. Healers start out with lower waiting times, but between the population of healers increasing at level 30, and the lack of people running lower level content, those queue times rise pretty sharply in the mid levels.