The correct sentence is the first one "I am a student and I also give classes". That is because you usually use "also" in front of a verb. However, there are some cases in which the position of the word "also" may change. I shall provide you with an explanation of the usage of "also" and its alternating positions in sentences:
* Firstly, you should know that you use "also" when you are giving more information about something.
1) "Also" is usually used in front of a verb. If there is no auxiliary verb, you put 'also' immediately in front of the verb, unless the verb is 'be'.
E.g.: "I also began to be interested in sports".
E.g.: "They also helped me out".
2) If the verb is 'be', you put 'also' after it.
E.g.: "I was also an American".
3) If there is an auxiliary verb, you put 'also' after the auxiliary verb.
E.g.: "The symptoms of the illness were also described in the book".
4) If there is more than one auxiliary verb, you put 'also' after the first one.
E.g.: "We'll also be learning about healthy eating".
5) 'Also' is sometimes put at the beginning of a clause.
E.g.: "She is very intelligent. Also, she is kind!"
*** Be Careful!
Do not put 'also' at the end of a clause.
Hope that helps!!! :)
Should you have any further doubts, do not hesitate to contact me. I shall be pleased to help!