1. "cannot" is one word, not two. There are cases where "can not" are used as separate words though (e.g. You can not only sing, but also jive.) In this case, "not only" is used with "but also" to describe two related abilities; "can" alone expresses ability.
2) It may be somewhat confusing to use "next weekend" and "following weekend", since the "following (next in time) weekend" may be the next weekend as well if you are writing the sentence this week. Perhaps you can use "the weekend after that".
3) "take a class" sounds rather off. Perhaps you can use "take/attend a lesson" or "attend a class".
"Since I cannot take a lesson next weekend, I will take *one on the weekend after that."
* We can use "one" to refer to a person or thing previously mentioned.
I hope this helps.