No it will not go bad in plastic. You do want to use food grade plastics, but if it had consumable goods in it, you are good to go.
I use a plastic bucket with sealed lid and air lock as my primary and its good for 2 plus weeks in there!
Try to do ur best to not airate your beer when transfering because it will cause off flavors and infect your beer. Don't freak out if you get a few bubbles or splash some, you will get better at it the more you do it.
Most important of all! Relax, don't worry, have a homebrew! (RDWHHB)