I can tell you as someone you just graduated with a B.S. in Meteorology and just got a job that your last sentence couldn't be further from the truth. Your options with a Bachelor's are limited but there are plenty of companies in the private sector that hire people straight out of college.
There are many different pathways you could go down if you want to pursue a Meteorology degree, but if you want to go the straight up forecasting route, you need to be aware that you will likely be working overnights, weekends and certain holidays. The pay also will likely be below your expectations. The work can also be pretty tedious at times. I would recommend going into straight up operational forecasting if that's your passion, but there are more lucrative opportunities elsewhere. Without disclosing too much information, I will be working in the private sector for a company that deals with weather risk management by combining my Meteo degree with a business minor. Going this route or supplementing with a programming background will open up many more doors aside from being just a forecaster.
So my advice going into college will be to see if you can handle the math and the physics because they are intense and weed out a lot of people. Get involved in extracurriculars in college and get at least one internship while you're in college. Along with supplementing a meteo degree with another skill, this will make you far more marketable without needing to go to grad school. Going to grad school is only necessary if you want to do research or work for the government.