How soon after university should I apply to grad schools ?
Good question, and it depends on a lot of factors, but my advice would be that having some experience in the world outside of college or university life is always a bonus when applying to grad school in any field. Getting a job, paying rent, saving money, paying down loans, meeting people from different walks of life, are all good experiences, even if they an be a struggle at times.
I had quite a bit of time between my graduation from college and my grad school, and I was a lot more mature and I was really committed to my future as an art teacher when I got to grad school. When I graduated from college, I was pretty unsure of what I wanted to do, I knew it would be somewhere in the fine arts field, but "where" was a question. You might have a more focused ideas than I did, but I still think having some time away from studies would be good, even if it is only a year or two. I can't think of a school that would not support your decision to take time off from studying to get some experiences outside of academia. I'm sure they would understand the economics of wanting to pay off some student loans. However, it may be there are some graduate schools in certain fields that might want you to apply right out of your undergraduate school, so to be safe, I might also run this question by your professors at your university.
You didn't say what kind of grad school you are looking to attend, but it would be a bonus to get some sort of job that is related to your field of interest. For example, a friend of mine has a daughter who wants to be a physical therapist and, while she is taking classes and applying to schools, she has a job working for a physical therapist as an assistant/technician. She is getting some real life experience, paying her bills and loans, and she should get a good reference from her boss. You might look into doing something related to your field.
I hope this helps; best wishes on your future.
Ellen is right, it depends on a lot of factors, but if you have the opportunity to work in the field you want to go into it will help your decision. Also if you go right into grad school it helps defer your loans.
Good luck and congratulations about thinking about furthering your education.