There are many methods of tatting. There is no one way that is right for all tatters. Each tatter needs to try out the various styles and find which is best for them. I suspect that finger tatting is the method to try first. Remember that a shuttle, needle, or spool is just a convenient way to have extra thread at hand for tatting. None of them are essential in order to actually tat. Finger tatting requires only thread and hand(s.) The tail end of the thread is held while the normal over/unders are worked.
The two-bladed tatting shuttle comes in many materials, wood, metal, plastic, ivory, bone, and styles, plain, painted, bejeweled and enameled. The use of a shuttle became increasingly popular in the late 19th and early 20th centuries. Perhaps because the fast flying ornamented shuttles were a way to also show off graceful fingers? The shuttle could be used alone, attached to a ball, or wound together with second shuttle.
The flat shuttle, made mostly from wood, metal and plastic was also popular in the early 20th century. The flat shuttle could be over wound with thread and thus eliminate many knots. Flat shuttles are also excellent when working with lots of beads as there is no "point" to pull the beads through which might possibly break or damage the beads. The use of the flat shuttle, however, means a slight change in hand position for working. The second half stitch is worked as normal. The first half stitch, though, works better if the thread is laid over the left hand instead of the right as is traditional. The change in method was called the "Modern Priscilla" method.
Also popular in the late 1800's was crochet-tatting. This method used a bullion hook. Similar to a crochet hook, the bullion hook had a shank which was the same diameter from handle to hook. Crochet-tatting enjoyed a resurgence in the late 1900's as well. Here the thread was wrapped onto the shank and the hook was used to pull the thread through stitches and form rings.
In 1917, Ms. Rozelle published a method of needle tatting. It also had the thread wrapped onto the needle. To form a ring, the needle was pushed through a previous stitch closing the ring much in the manner of the self-closing mock ring today. However, the method of joining was not the modern method. Here the needle was simply pushed through the opening of a picot and more thread wrapped on.
This method of needle tatting is not to be confused with the method of using a needle as if it were a shuttle. The needle is attached to the thread and then manipulated in the same manner as a shuttle. There are early patterns which are all one shuttle work, apparently, but the bare thread space between the rings has threads crossed around it or even "faux" double stitches applied over the bare thread in a blanket stitch fashion. These small spaces are too small to allow a shuttle to pass through, so a needle used as a shuttle is the most obvious method to create that effect.
And today there is the modern needle tatting method. Here the needle can be used just with a cut thread or attached to a ball for larger pieces. The thread is wrapped onto the shank of the needle and slides off onto a carrying cord or foundation cord. Rings are formed by pushing the needle through a thread opening. This type of needle tatting allows for an expensive thread, perhaps metallic, textured or fluffy, to be wrapped and carried on a less expensive plain thread.
In the late 1970's, Toshiko Takashima, developed a method of Japanese needle hook tatting. Here the needle has a shank which is the same diameter from end to end but each end has a hook on it which is used to form the rings and chains. It is similar to the older crochet-tatting style. The instructions for this method are posted for public use.