This is a good question, but it will have at least as many correct answers as there are math professors. I think you would do well to find a math professor that you feel would be willing to give you a decently full, honest answer and be responsive to any followup questions you may have. I think this is a perfectly appropriate question for any student to ask me at any time: after all, by teaching mathematics at the university level I am implicitly promoting that profession, but my promotion of that is not absolute and unqualified: some students should seek this as a career and others should probably not.
Short answer from me: the amount of time I spend teaching undergraduates, teaching graduates, mentoring graduate students and doing my own research* varies tremendously from week to week and semester to semester, for instance because the number of courses that I need to teach in a given academic year is not a whole number. Also having a PhD student (and to a lesser but still certainly nontrivial extent, a master's student) is a fair amount of work on average, but there are times when it becomes a tremendous amount of work: e.g. for both the master's student and the PhD student whose theses I successfully directed, there was about one semester where the amount of time I was giving to them was in excess of any one of the classes I was teaching. And I think that the amount of time I devote to my students is probably average to slightly below average (I tend to encourage independent work in students rather than weekly reassurance), so when I see colleagues who are supervising four students and meeting with each of them every week, I feel deep admiration and wonder how they manage to get anything else done.
My research goal is to submit roughly three papers per year: less if they are long and substantial, more if they are short and light. Generally I spend more time doing research over the summer than during the academic year. Sometimes I have spent entire semesters doing only "caretaking" of my research program: i.e., performing routine revisions on papers, and so forth. More recently I have tried to do at least some mathematical research every week...although it may not be the research that I am "supposed" to be doing.
I would classify being a professor as a highly competitive occupation. It is competitive in different ways at different points in your career -- there are times when it is so competitive that one thinks twice about staying in the profession but rides it out in expectation of future, better times -- but I would say that for a professor at a research university the competitive aspect is always there. For instance we compete for higher academic ranks, for raises, for status within one's department and in one's profession, for both internal and external grants, and so forth. Much of this competition is negative competition that must be dealt with, I would say. (It would be nice if I didn't have to show myself as being good enough to get a job offer at a different university in order to get a decent raise, but that seems increasingly to be the reality as time goes on.) On the other hand mathematics is also inherently competitive in a more positive way: we always try to push the boundary of what is known, to improve on past results, and even more basically to improve one's own knowledge, technique and skills. This kind of competition is much of what keeps brilliant people doing mathematics over the course of their long life, when by every external measure they have achieved essentially maximal career success.
*: This is not an exhaustive list of my professional responsibilities. But I don't want to get into more detail than this right now.