Well, it doesn't quite work that way - you won't "double major in nuclear and astro". My first advice to you is to always keep your options open - you never know how much you really like a subject until you actually get your hands dirty!
With that warning out of the way: assuming you decide to go into physics, you should find a school with a good department, to maximize your opportunities. If you're still in HS, you shouldn't focus on such specifics as "nuclear" vs any other form of physics - just go someplace where they are doing good research. See what courses they offer (usually the high numbers are the specialty courses) - that will give you a good idea.
When you decide to major in physics, you will take all the standard courses, such as quantum mechanics, E&M, statistical mechanics, etc. Then most places have you take a few specialty courses, and you can also do a senior thesis and do independent research with a professor. It's not until you get to grad school that you would specialize as a "nuclear physicist" or a "astrophysicist"; in undergraduate, you're just a physicist! ;-)
I got to know my academic adviser on a first name basis, and then one day of my sophomore year, I asked if he knew of any professor looking to hire an undergraduate, and he said, "Yes, come to my office first thing Monday morning." And that was that! So when you get to your school, I would recommend trying to get in with a professor or two, and then take it from there.
I would also recommend trying to get a degree in "Physics and Astronomy" rather than just "Astronomy" - most schools do it this way, but some don't. I think you get a much broader education that way, especially if you're not sure you want to become an astrophysicist. I might make a few enemies with that advice, but there you are. ;-)
Anyway, I hope that answers some of your questions. Good luck!