Puppies investigate the world through their mouths, for us it can be very painful and annoying to have sharp puppy teeth grab your hands/arms/legs every two seconds but biting is a normal behavior in puppies. For them, if something is within reach, it will probably be picked up and chewed, if it moves fast, it will definitely get bitten. What they are doing is simply trying to elicit play. Play is by far the best way to bond with your pup and is a great way to reward him during training.
How to play without getting bitten- Use tug toys that he can bite. Old knotted towels/shirts or a favorite toy with string attached will work. You need to encourage him to bite one end of the toy while you hold the other end. Ensure your tug toys are long enough and soft enough for your puppy to bite. Your toy should be able to touch the floor while you are holding the other end. This allows you to keep the game low to the ground and not encourage jumping up. It also puts distance between teeth and hands.
Start by animating the toy on the floor and saying “getit” every time your pup grabs the toy. You hold on to the toy and let him grab it and shake it. Let go of the toy sometimes so that puppy is encouraged to come back to you to get you to start the game again.
Also teach a word for letting go. To do this you simply stop the game by putting a finger in puppies collar and keeping hold of the toy, release the pressure on the toy so that it becomes boring. You can use a cue like “drop it” or as soon as puppy lets go say “thank you” and immediately invite him to grab it again with a “getit”. He will quickly learn to let go when you stop playing in order for the game to start again and eventually the word “drop it” or “thank you” will become his cue to let go.
Keep toys out of your pups reach while they are not being played with. It will keep the pup interested and more likely to want to play with them when given the opportunity. Plant toys around the house (out of puppies reach) so you have them easily accessible and as much as possible, take the game outside.
Once your puppy is getting the idea of the game then you can start to add in a “sit” “are you ready” “wait” before the “getit” and soon you will have a dog sitting and waiting patiently for the game to start.
Do not play with your puppy unless you have a toy for him to grab. Don't let anyone in the house roughhouse with him or roll about on the floor with him. If a puppy starts biting your hands/body immediately stop the game and put the puppy up. They will soon learn that you will stop playing with them if they bite.