What you'll need to do is get a
hydrometer. I have one by Instant Ocean. They aren't very expensive.

There are other brands too.
Then you just put a bit of salt in, mix it in well so that it's all dissolved, then take a reading. Too low, add a bit more salt and repeat, too high, dilute with water. You'll get a hang of learning how much you need to pour in. Oh, but don't pour the salt directly into the tank. Do it in a bucket when you do water changes.
Now, if the tank he is in now is fully cycled, what I suggest you do is move him and all his filter media, decor, and substrate (if you plan on using a different substrate in the big tank, <I suggest something with aragonite> then just put the substrate in a mesh bag/pantyhose and hang it in the tank) directly over to the larger tank. Then add salt to make it 1.002
SG. When I got my puffers, I added a bit of marine salt to their water and they instantly looked healthier and lost most of the grey on their tummies. After that you can raise it by up to .002 SG each week. Also, keep a close eye on the water parameters, ie
ammonia,
nitrite,
nitrate.
I hope I answered all your questions....