Throughout my life I have thought that the correct expression is to take revenge; however, I have also heard people saying I'll revenge. Which is correct?
Revenge is most commonly used as a noun, e.g. "I vow to take revenge on him for stealing my girlfriend." The words "to take" are optional, and you could just as easily say: "I vow revenge...". You can also substitute "upon" for "on", but it sounds more pompous.
Revenge is less commonly used as a verb. The construction is always in a very specific form, from what I've seen. Usage: "I will revenge myself on him in the most bloody way!" Note the use of the first person reflexive pronoun, the preposition "on" (which can be replaced with "upon") and an object (the party you are going to wreak vengeance upon). Note that this usage would sound highly affected and pompous to most speakers, and should be reserved for when you're going for the "grand effect", e.g. in a bombastic literary work.
Contrast this to avenge, which is always a verb, and vengeance, which is always a noun.
I believe what you have heard is an eggcorn. It's simply someone who has misheard and reused the phrase "I'll avenge".