It may help to treat the pipe the way you treat a thermometer. You stick it in your mouth and just hold it there, breathing naturally through your nose and swallowing spit from time to time.
The smoke will waft into your mouth on its own. When it does, just take the pipe out of your mouth and blow the smoke out of your mouth. In time, you'll get comfortable enough with this to release the smoke from your mouth without taking the pipe out of your mouth (or to push the smoke back through the pipe if you want to). It's like holding a lit cigarette in your mouth while working on something with your hands.
If the pipe is going out, give it a "cigar puff" like you mentioned before. Then resume treating it like a thermometer.
Don't worry about doing it the "right way," just focus on calm, natural breathing through the nose. The pipe will take care of the rest. In fact, you'll find it puffing itself so fast that you have to stop it by letting it sit alone for a few seconds.