First and foremost, if you haven’t taken an MSF beginning rider’s course, you need to do that pronto. That will answer all of your questions and vastly improve your riding ability and preparedness for street riding.
When you say “downshifting” do you mean a single downshift, or downshifting all the way when stopping for, say, a light?
Either way, it’s better, when possible, to engage each gear as you go down for a number of reasons. First, it’s the easiest to do smoothly once you’re a proficient downshifter. You don’t have to guess as to how hard to rev the engine and how hard to brake and when to let out the clutch to make up for the big jump from a high gear to a low gear.
The trick to smooth downshifting is to be very fast and to blip the throttle. Being fast will develop with time and repetition. Blipping takes some practice. To downshift, you pull in the clutch, bump down a gear, and let the clutch back out. To blip, what you’re going to do is give the throttle a twist while clutched in to get revs higher for the lower gear. One big and quick rev. Don’t linger. Open and close it immediately. You have to be ready to clutch back in at the “top” of the blip, so at first you might want to wait to blip until you’re in the lower gear, clutch still disengaged. As you get more comfortable, you’ll blip as you downshift for extra speed.
One thing people sometimes do is wait too long to clutch back in. You have to catch the blip when the engine is revved up or you don’t get any benefit. It can be a bit worrisome at first if you’re afraid you revved too hard and you hesitate, but you just have to go for it. Start with a smoother change like 4th to 3rd (on a 250) or 3rd to 2nd (on a bigger bike). 1st gear is the hardest to hit smoothly.
But yeah, emphasis on taking that BRC before you ride more!