As with so many important decisions in our lives, there really isn’t a right or wrong answer to this question. We can learn from our choices and how they turn out regardless of which choice we make in any situation. Practically speaking, the process for applying to graduate school begins early in the year and you may have already missed the deadline for starting a graduate program in the fall. Still, it is worth looking into because it may not be too late, especially if you are planning to go to the same college you went to for your undergraduate education.
If you have missed the deadline, you will have the opportunity to work for 6 or 7 months, hopefully in the field you intend to go into. If you still have a chance to be considered for graduate school admission, I would suggest going forward with your application. That way you cover your bases in case you decide graduate school is the way to go and if you decide to work for a while, nothing is lost by applying.
Student loan debt is a huge problem for most students these days. You are right to be concerned about coming out of college already owing money and then needing to finance your graduate education. The truth is that for most students just graduating from university, it is hard to find a job that pays well enough to cover your living expenses and service your student loan debt.
If you are serious about your career goals and know where you want to be professionally, I think it makes sense to continue your graduate education. It will put you into a position to get a higher paying job after completing your graduate degree. In addition, some student loans can be put into deferment status as long as you are a full-time student, thereby allowing you to be in a better position to service your debt after you graduate.
Naturally it makes sense to borrow as little as possible to finance your graduate education. Look into grants, scholarships and graduate research and teaching assistantships. These can greatly reduce your out of pocket expenses. If possible, try to work part-time rather than rely exclusively on borrowing money. The less debt you graduate with the happier you are going to be.
Working and paying off prior student loan debt is certainly another valid choice, especially if you can gain work or volunteer experience in your chosen field. Psychologically, either one can be beneficial and both paths can lead to positive outcomes, but if you are clear on your career goals you might be happier pressing forward and moving more quickly toward your goals.