Back when my friends and I were first getting into Magic, we didn’t know how priority worked. You’d commonly hear “too late, I already ended my turn” in response to people wanting to cast spells at EOT, or “Too late, it already resolved” in response to an attempted counterspell. But as we know now, you can’t do that. Magic doesn’t work like that; for any given thing on the stack to resolve, every player gets at least one shot at responding before it can. For the game to move to a new step/phase, each player gets at least one chance to do things beforehand.

So, knowing that, you can’t ‘rush’ your opponent past things. You can’t untap and immediately draw a card to try and rush past the upkeep [whether your intent is to make them ‘lose’ their upkeep trigger, or just to keep them from casting a spell at that time]. You can’t jump straight to declaring attackers [or blockers, if you’re NAP] without letting them do stuff beforehand. Your opponent is going to get priority at least once per step, every step, so just let them. It’s a lot easier to do things by the book than it is to try and skirt the rules that you’re already on the wrong side of and then have a Judge come sort it out.