Hi there!
Your Border Collie puppy must be really cute when he is not biting you <g>
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It is a great idea that you enrolled him for a puppy training class.
I have rescued and fostered many dogs and they behave fairly well first 2-3 days because they are still not sure if this will be their permanent home or if you are a nice and trusty person, etc.
I call this period as "Bud" and they start "Blooming" as they get used to you and other animals in your household, etc. This is the time you really need to make sure your new dog understand various rules in your family.
Therefore, it sounds like your puppy is starting to bloom and trying to figure out how far he can go, as he is getting more and more comfortable with you. It is important time to start building trustworthy relationship with your new dog. Thus, the training class is a very good idea.
As for biting, I would tell him it is not acceptable from the very beginning. You can say "Ouch!" and give him something to bite right away (chew bones, Kongs, etc.). This will tell him what he can bite but what he cannot. Puppies like to play using mouth among siblings and learn how hard they can bite each other. However, we, human beings, don't have a furry skin so he should learn it is not acceptable to put his teech on our bare skin. If he does not listen to you, you might want to use a spray bottle (w/water) to get his attention.
Good luck!
