Difficulty: Easy
If you have an especially cute puppy (and what puppy isn't cute?), visitors think that it is absolutely adorable to be introduced with a tiny paw-shake just as if to say, "Hi, nice to meet you! Why yes, it is okay for you to rub my belly now!"
Step 1: Have your dog sit in front of you.
Step 2: When he lifts his paw click and treat.
Step 3: Repeat five times.
Step 4: Wait for him to offer lifting his paw (however slight). Click and treat.
Step 5: Say the word "paw" each time he lifts his paw. Click and treat.
Step 6: Say the word "paw" and click and treat when he lifts his paw.
Step 7: Modify the trick by rewarding only higher lifts. You can encourage him to lift his paw by tickling the hollow behind his paw.
OUR EXPERIENCE
When Caspian learned "Paw," I started out by having him sit. I waited for him to lift his paw. As soon as he did lift it, I clicked and treated. I waited again, and clicked and treated when he lifted his paw. It was OK that he was only lifting it up a little bit. I wanted to capture the good behavior, however slight it was. After a few times repeating this, Caspian figured out that he got a treat whenever he lifted his paw. He started doing it more frequently, and I started saying the word "Paw" whenever he lifted it. Soon, when I gave him the command, he lifted it! In other training sessions I was able to shape the behavior so that he lifted his paw higher and eventually placed it in my hand.
TEACHING TROUBLE
He will lift his paw, but only a little bit! How can I get him to lift it higher?
Click and treat at the beginning, no matter how high he lifts his paw, to encourage good behavior. Then, get tougher! Start to only click and treat when your dog lifts his paw higher, and the higher he lifts it, the bigger treat you give him. He may get frustrated, but he'll get it soon enough.
Tip: "You can encourage him to lift his paw by tickling the hollow behind his paw."