During your turn, if the stack is empty and you're not moving to another phase, then you and only you have priority. This means that you are the only person that can do anything. When you go to cast a spell, naming it, revealing it, and paying for it all happen (I forget the specific order, though).
Once all of that happens, you can either hold priority to do something else, or you can pass priority. Only then can someone respond to what you've done, but not to prevent you from doing it, except by countering it. Because of that, no one can ever remove your ability to cast a spell or activate an ability before you have a chance to do so when you can legally do so.
Edit: fixed some stuff; I wrote the original response on my phone, hence the errors.
Edit 2: if you want to move to your next phase, then there is a passing of priority before that happens; this is usually done by way of a shortcut, e.g. by saying, "Move to combat/second phase/end step," at which point your opponents can either do nothing or respond by doing something before moving to the proposed phase because they have been given priority.