Roughly, because I'm at work and REALLY should be finishing a couple of clients' tax returns...
It starts well before formal leash training, with attention work and "follow me" games so the puppy establishes a habit of staying with you. (Note - I'll use c/t to indicate click/treat, because it is what I am most used to, with the caveats that clicking is not universal AND that while treat are very useful at the start as rewards, they are phased out of common use pretty quickly to avoid the puppy simply being lured around by the food.)
So you'd start with short sessions, verbally encouraging the pup to follow along and stay with you, c/t for being in the correct position and maintaining eye contact (click while the pup is moving forward with eye contact, then stop and drop the treat, then start again with verbal encouragement - this links the click (marker) with what the pup is DOING, not with the treat). As you're just starting, your progression forward will be limited. As the pup learns he's getting clicked for being in heel position as you move, you can extend the distance and introduce turns and halts - I start this very gradually, not expecting 90 degree corners until he gets the idea that turns are going to happen and he has to watch for them!
Since I taught Ozy "touch" very early on, I tend to use my left hand as an added stimulus - sweeping forward as we begin, then holding it at my waist as we walk, sweeping upward into the "sit" cue as we halt.
Once you're walking "out in the world", even a well-trained pup is going to give pulling a try when there's something REALLY EXCITING ahead of you. For this, my trainer uses the "be a tree" method. As soon as the pup goes ahead, you stop. Completely. Don't let your hands follow forward with the leash, don't pull back on it. The pup will lunge around a bit wondering why he's not getting anywhere and will eventually look back to see what happened to you. Click the look. And since he knows that the click means he gets a treat, he'll come get it. Back up quickly a few steps and encourage him to close the distance. If he's really into whatever has caught his attention and keeping the leash taut, begin to back up so the leash tension increases. NOTE - you are not pulling him toward you, you are backing up. He only gets the "correct" marker for turning toward you and only gets rewarded for actually coming to you. Then start forward again - or start going in the other direction. You'll probably have to repeat the "be a tree" pattern a couple times each walk, but pup will quickly learn that lunging toward whatever he wants only gets him "penalty yards" in the other direction. Intersperse with attention work, halts and turns to keep his attention on you.
Another note - I don't have any moral objections about letting a dog who is well into his "heel" training bang himself into his (flat) collar by a short stop or quick turn if he's really not focusing. But I only use flat (buckle or martigale) collars, and I don't use this method while the dog is still learning what I mean by "heel".
That's my read on "positive training", really - I don't train the pup by physically correcting his mistakes, I train by rewarding his correct behaviors. Sometimes (mostly when just starting out) that means treats, sometimes it's by withholding what he wants (as in - you will make progress toward the REALLY INTERESTING thing only when you stay at my side, never while you're pulling), sometimes it's play (I use this for "stay" work a lot, because it makes a clear break between the obedience and the reward). My experience with the leash correction methods has been that you're essentially punishing the pup for not understanding what you're asking, and that seems wrong to me. Now my older dog will sometimes give me the "doggy finger" when I tell her to quiet or something like that, and I have NO problem reading her the riot act and putting her in a good long down-stay right in front of me. I holler, I smack my hand on the table/counter, I stand over her and shake a finger, but I do not physically touch her unless she's actually in a dangerous situation. But Ozy is just a puppy and is just learning to translate our weird language and requests, so I try to set him up for success and just redirect when he guesses wrong.
I hope this has been clear and helpful - now I really have to get back to those tax returns!!