You always attach 도 to a noun, so if you want to use it with a verb, you have to use a noun form of that verb.
Also, depending on which word you attach 도 to, you emphasize that the "also/too" refers to the specific word.
For instance, I feel like "저는 영어를 가르쳐요" and "저는 영어를 가르치기도 해요" mean exactly the same thing.
These sentences are not the same.
The sentence without 도
저는 영어를 가르쳐요
I teach English
A plain sentence without implying that this is an addition to something that was said previously.
Adding 도 to a nominalized verb
저는 영어를 가르치기도 해요
I am also teaching English
What this sentences really means is "Teaching english is also something I do". Maybe you said something before about what you do with english, like rapping in English or studying English linguistics, but you want say, that you do this other thing with the object as well.
Adding 도 to the subject/topic
저도 영어를 가르쳐요
I too each English
In this situation you are saying that you yourself do something, too. Maybe someone said to you "I teach English", than this would be the right expression.
Adding 도 to a the object
저는 영어도 가르쳐요
I also teach English
Here you may have stated before, that you teach french or anything else, but you do the teaching also for another thing, in this case English. “I teach French. I also teach English.”