There is more than one "correct" way of gathering and cooking tree sap for syrup making.
Shallow tapholes penetrate the sapwood of the tree and usually produce a light-colored, delicately flavored syrup. Deeper holes, on the other hand, enter the heartwood and typically yield a darker, more strongly flavored product. In either case, remember that you'll need warm days and subfreezing nights to make your trees' sap rise and fall enough for you to "get your share."
Either boiling or low-heat steaming will convert collected sap to syrup. Each method has its advantages and disadvantages. While low-heat evaporation is very time consuming, boiling sap requires constant attention and stirring because it tends to bubble over without warning and is very flammable. (it's also more difficult to "stop" hot syrup at a precise consistency when it's rapidly boiled rather than slowly steamed down. "Proper" syrup should keep at least a year at room temperature; if a batch turns rubbery, it may have been slightly overcooked.)
Now that you know several ways to make syrup, just pick the methods that seem best for you. After all, once you've cooked up a gallon or two of your own sweetener you'll probably have developed some "pet" techniques of your own!
(Additional information about the production of maple syrup can be found in "How to Make and Market Maple Syrup," and "How to Make Maple Syrup at Home.")