What is the ruling on saying "I make a vow to do such and such" without specifying the utterance, and does this fall under major shirk, knowing that the speaker did not know the meaning of a lawful vow and that it is for Allah alone?

A vow (Nadhr) is when a Muslim obligates himself to an act of obedience to Allah Almighty that is not otherwise legally (shar‘an) binding upon him. It is established by any utterance that signifies commitment, such as saying: "I have vowed to Allah." If its intention is to draw closer to Allah (such as prayer or fasting), then it is obligatory to fulfill it. However, if it is for something other than an act of devotion (such as a permissible act), then you have the choice between fulfilling it or abandoning it with an expiation for breaking an oath (kaffarat yamin): feeding ten poor people, or clothing them, or freeing a slave, or fasting three days if you cannot afford the former options. As for vowing to other than Allah, it is polytheism (shirk), and you must offer sincere repentance for it.