I think if you're an independent you have the choice of voting in either the Republican primary or the Democratic primary. But only one
it depends on your state. Some have strictly closed primaries (like NY which is why I didn't ask PB), some have open, some have semi open.
In some where you're allowed to vote without being affiliated the primary you vote in affiliates you until the next election,in others it doesn't.
Edited to add- thank you Olivant for the Texas info.
I don't get to vote in primaries, I'm a former NY resident and now I live in NJ. Both are strictly closed. Although in national elections I tend to vote third party anyway, as neither state is normally a battleground and it's my little way of saying F you to the two party system.