We don't normally use "revenge" as a verb. The word "avenge" means something like "to get revenge for" (or maybe "to get revenge on behalf of" would be better). "I'll avenge you" is the way to say it.

Hello! "I will avenge you" means that I will take revenge on your behalf - if someone has done something bad to you I will do something bad or harmful to that person for you. "I will revenge you" should rather be said in this way: "I will take revenge on you" and that means the person (I) will take revenge directly on you and not on behalf of you as in the first sentence.

Avenge is the only form of this verb. Revenge is a noun and a form or word "vengeance". You could say, for example, "I will avenge you, Sensei, and that will be a glorious revenge". "Revenge" must never be used as a verb. P.S. I once a Brit guy who reckoned that revenge and avenge were two verbs with opposite meanings, he later got fired from his teaching job because his High School certificate was a fake one.