Congrats on your new GCC! They are lovely little birds!
Like any new animal, give them up to a week, week and a half maybe (or longer, just go at your bird's pace), to adjust to their surroundings and acclimate. Moving from a store can be very scary and stressful, causing the bird to defensively lash out. I don't know your home life or what the store was like, but any other pets, people, general surroundings, could be all very new and very scary. Birds, especially this small, can be prey animals and will react accordingly. I was lucky in the sense that Callum was immediately puffed up, cuddling, and sleeping against my face and neck the day I brought him home.
Avoid feeding treats or food through the cage bars, as this will encourage your bird to bite every time something comes in between the bars. With my GCC, Callum if he's getting a treat, I usually open the door and offer him the treat instead. But, since he has free roam of my home when I'm at home, he usually just flies over to me if he sees a treat coming lol. Either way, door opening means good things only-- a treat, being handled, or allowed to have free time.
If you would like to work with your new family member on step up but he remains nippy, try using a stick for stepping up. Have favorite treats on hand and immediately reward when he does step up on the stick. Don't flinch when he attempts to bite, as you flinching away just encourages the behavior ("If I bite, the hand leaves me alone.")
As others have mentioned, coming back in the room (whether to reprimand or just do human things) following a screech, you're encouraging the screaming behavior. Birds are exactly like human children-- any attention, even negative, is attention. Usually, if Callum's screeching, I'll wait outside the room, out of his sight, and wait until he's finished, and then walk in. Then if he remains quiet, I'll take him out.