I use "heel" as a formal, precise position. Both stationary and moving. I use "with me" for just walking on a loose leash. I use the heel command to get my dog into heel position as well as for walking at heel. I only ask for heeling if we're doing a formal training session or if we're in a crowded area where I don't want my dog to have the freedom that a "with me" allows.
I teach it primarily by muscle memory. I lure it. A lot. Until it pretty much becomes the default place for the dog to be. There are down sides to doing it that way...it can be difficult to get rid of the lure, it can make it difficult to train sit fronts.
I was also wondering what common obedience ring commands and signals were.
I don't know about common, but I use "heel" for the dog to move with me at heel, I use "wait" when I'm leaving him for a stay or asking him to remain sitting while I throw the dumbbell
I use "sit" "down" "stand" as needed for those positions. I use "come" for a recall. "Bring" for retrieves. "Over" for jumping. For finishes, I use "heel" for a flip finish and "bye" for an around finish. "Run" for a send out.
Some people use "wait" when they leave the dog for a recall or to throw the dumbbell and "stay" for the group stays/stand for exam. Some people use "come" for the first call in drop on recall and "front" to call the dog to a sit front. Some people use "over" for the retrieve over high - I think that varies depending on the dog -- if he's more focused on getting the dumbbell than the jump vs. if he's more focused on the jump. Some people use "high" and "bar" for the directed jumping.
For signals, I only use them in utility for the signal exercise. The heel signal is kind of a forward sweep of my hand and my thigh...also stepping off on my left foot, which is really all the dog needs, but I do the hand signal so the judge knows I cued him.
For the stand at heel, I use my right hand in front of his face, when I leave him on the stand, I use my left hand (again not necessary - stepping off with my right foot is all he needs). For the drop, I raise my hand straight up and then bring it straight down. For the sit, I hold my hand down and about 30-45 degrees away from my body and bring my lower arm up, bending my elbow and also bending my wrist so that my palm is facing the ceiling. For recall, I bring my arm out up parallel to the ground and almost directly in front of me (maybe a slight offset) and then bring my hand to my chest in sort of an arc. For the finish, right or left hand at my waist sweeping from the front to the back.
Hope that made some sense.