Reddit reviews Weather Analysis and Forecasting Handbook

Your current major is a fine major to have if you want to go to grad school in the area of atmospheric science. My degree is technically "geoscience" and I only have a concentration in operational meteorology, and I got into grad school no problem.

While thinking about grad school you also need to think about what you want to do. Do you want to forecast, research, teach, or something totally different?

If you want to forecast, your program is perfectly suitable for that. If you want to learn even more about forecasting, I highly recommend this book (Weather Analysis and Forecasting Handbook by Time Vasquez). And like anything else you have to practice, practice, practice to be good at it.
You can also get more experience forecasting by calling up any NWS offices within a reasonable distance and asking the MIC if you could shadow a forecaster on weekends. I have had several friends that have had success in doing this, and they got tons of great hands on experience. Or if you are not like a lot of us weather/atmo geeks, that is you aren't overly shy or too self conscious, consider getting a second major in communications and get a summer internship with a broadcast met at a local news station. Most of my class mates were broadcast mets, and they have become fine forecasters and get to do it everyday and make loads more money than I do!
Forecasting jobs can be difficult to come by however. There are usually a few broadcasting jobs available, but it might mean that you have to move to the middle of no where Wyoming. There are also various private sector jobs, but they almost always want years of experience.
You are probably more likely to win the lotto than get a job with the NWS under our current economic situation, however, I know one person who got a masters in GIS and landed a job. I have been told by him that currently they are looking for people interested in meteorology but that also have a strong background in GIS.

Or if you want to do research and or teach, than grad school is definitively your option. First you need to decide what your research interests are. Do you want to do numerical modeling, study cloud physics, atmospheric dynamics, atmospheric chemistry, etc. Then start looking to see if you could get a second major, or even a minor in an area that would help you stand out in the stack of applicants. So, for example, if you are interested in modeling, than maybe consider a second major or minor in computer science. After that start looking at schools and professors that do similar research that fits your interests best. OU is a very competitive school, but for good reasons. There are many other equally good, and some maybe even better depending on what you want to do, than OU. So I suggest you take your time looking into others. Once you find a person who you would like to work with, email them and see if they know if they will have funding for a new student the next year (do this during the fall of your senior year).
Then once you've figured all of that out start applying. Remember, find a way to make your self stand out by finding a second major similar to what you want to research, and/or helping one of your professors with their research, and/or by getting into an REU. Also take the GRE seriously. Often times programs have some cut off score where they wont look at any applicants with a score below it. I know that in my program the cut off was 1200. I'm not sure what the equates to on the new scale though. Getting into grad school in atmo science is very competitive. Again my program took in 10% of the applicants, and there were 150 applicants. So making yourself stand out is key.

Whew... I got really long winded there. I know that I gave more info than what you were asking for, but I remember being in your shoes and these are some things that I wish people had told me before I started the application process. Another thing to do is to talk to your adviser about your goals and get advice from them too.
Best of luck!