Congratulations on your new baby! Its important, once he's big enough, to let him gradually get used to physical activity, so he doesn't get too excited about doing something fun and overexert himself. Babies can be damaged by stuff that, for an adult chin, wouldn't be as bad (flying off of a spinning wheel, a fall from a shelf, etc.). That's part of why the suggestions for no playtime came up. As safe as we try to make it, out of cage time always has hazards, and chinchilla bones aren't much bigger than toothpicks. They're fragile little critters.
Handling him in his cage is fine, just keep an eye out for any places he could drop all the way from the top of the cage to the bottom. A couple of people in the Classified Ads thread make hammocks that will stretch through most of the cage to fill in gaps between shelves etc., or you can make them yourself if you're crafty. And fleece is generally safe for chins of all ages, so he'll have something to cuddle with when you're not around.
Don't flinch or jump if you feel him nibble on you a little; that's how they show affection for each other, by grooming, and if you startle, he'll be less likely to do it. Being calm and quiet and moving slowly seems to have helped me the most with gentling mine. I also sing to my chins, so don't feel like a dork for just talking to him. When you've got video of yourself singing to half-asleep chinchillas lurking on your SD card (guilty 0/) you can go ahead and feel like a dork.
It's going to be so much fun for you to see his individual personality come out over the next few weeks. Enjoy it!