Well let's see, that was a lot of questions.
First my background. Have a BS in math, then went to a Air Force recruiter and asked for a technical job. Went to the Navy's grad school for a year to get the basics of meteorology and am now a weather officer. I had ZERO prior interest in weather but found that it lined up with a lot of my interests.
So the question is, grad school in meteorology. Yes it's possible without a BS in meteorology. A lot schools have no problem with math and physics undergrads and will also accept engineering degrees. Which classes would help you from a physics major? Thermodynamics for one. Also any sort of fluid dynamics. Electromagnetics will give you a jump on truly understanding radar and remote sensing techniques. Numerical analysis? You bet that would help. Plus math, math, math, math!
Job prospects? I don't really know, I'm not planning on staying in weather when I get out next year. There's always the government: NWS, NOAA, the military, etc... although the AF is "force shaping" weather dudes pretty hard right now. Next are the government contractors. Finally you have industry. Apparently there are some weird weather jobs out there you wouldn't expect. A master's degree is pretty much required for a lot of these. A PhD places you in academia or government research.
That said, weather is a highly observational science. I did NOT understand that going into this deal. I did fine in school, but it was quite a shock when I got to my unit and realized the shear amount of data available throughout the normal day. Surface observations ever hour or more, upper air data twice a day, multispectral satellite data and derived products, radar, tons of models (is it reliable today?!), automated sensors vs. observers. Sure my math background meant I had good analytic skills, but coming out of school I barely knew what a cold front was let alone what kind of impacts to expect.
If weather has always interested you and you're a weather nerd with a physics degree, I'd suggest going straight to grad school. Otherwise, get a job and stare at data for a year before you go back.
GRE? There's a subject exam in physics and would probably help to take it if you could score awesome, otherwise forget about it and focus on the general exam. You should take them ideally over summer before senior year.
Websites? Do you need weather websites? Check out the American Meteorological Society... all their pre-2000 journal articles are free online. Also there are the COMET MetEd modules for learning about forecasting. Good luck and please ask more questions!