I want to teach songs (the songs are around 1 - 25 years old) I do not own to private students online. My concern is that if I charge a fee for these classes, would it fall into an illegal activity?
You can charge fees for such a service, unless there is a local business regulation preventing it. Depending on your location, it might be required that you receive a government license to engage in any business, so you would need such a license. I've never heard of music teaching being a regulated business, but it might happen. The more important question is whether you can perform the service, with or without fees.
Songs of that vintage are protected by copyright, meaning that you need the permission of the rights holder to make any copy of the work, or to perform it publicly. You might legally borrow a book of songs from a library and perform them for yourself or your family, which is legal. If you have an online student and they likewise borrow a copy of such a song book from the library, then the two of you would have legal copies of the songs, without owning them. It is not legal for you to just copy the book and send it to the student. It might be found legal for you to do limited copying, under the fair use doctrine, but then it would matter substantially whether you charge for this instruction (for-profit use disfavors "fair use").
One of the essential rights of the copyright holder is the exclusive right to authorize
in the case of literary, musical, dramatic, and choreographic works, pantomimes, and motion pictures and other audiovisual works, to perform the copyrighted work publicly
Perform means perform, i.e. singing the song, and to perform or display a work “publicly” means—
(1) to perform or display it at a place open to the public or at any place where a substantial number of persons outside of a normal circle of a family and its social acquaintances is gathered; or
(2) to transmit or otherwise communicate a performance or display of the work to a place specified by clause (1) or to the public, by means of any device or process, whether the members of the public capable of receiving the performance or display receive it in the same place or in separate places and at the same time or at different times.
The question is whether a one-on-one private instruction session (e.g. via Skype) is a public performance.
An internet instruction session is not a situation where "a normal circle of a family and its social acquaintances is gathered", so yo are not excused by that part of the law. The internet is "open to the public" in the same way that a concert is open to the public – you may have to pay to get in, but it is open to the public. However, in a 1-on-1 Skype session, there may not be a substantial number of persons gathered: the law does not say what a substantial number is. The second clause accommodates internet applications, and says basically that "substantial" can be cumulative, thus if you have 10 students, that probably is "substantial", but if you only have 1 student, that might not be substantial.
There are explicit legal permissions for instructors to perform in certain educational circumstances, whereby an instructor can have a class with a number of students. The rules for that permission are not simple. It allows performance
by instructors or pupils in the course of face-to-face teaching activities of a nonprofit educational institution
which this is not. The remaining paragraphs, especially 2 and 4, are very complicated (that's code for "nearly incomprehensible"), but do not seem to provide an exception for for-profit instructional performances: I would say that you really need to hire an attorney with expertise in music performance licensing law to get a definitive assessment. It is more likely than not that such a scheme, without a properly licensed copy of the songs, violates copyright law.