An adverb of manner expresses the manner in which an action is done. Now, if things can be done in a basic manner, they can presumably also be done in an expert manner. So if you can say "I'm searching basically for conversation classes" is an adverb of manner, you could also say "I'm searching expertly for conversation classes." I don't think this is what you mean.
You are probably intending this adverb to be a sentence-modifying adverb. Sentence-modifying adverbs are usually used at the start or end of clauses, or right before the verb:
"Basically, I'm searching for conversation classes."
"I'm searching for conversation classes, basically."
"I'm basically searching for conversation classes."
The position of the adverb can change the intended focus of the sentence. To put basically at the front highlights or emphasizes the idea that "what I'm about to say is a basic summary, etc. of what I'm doing or searching for." The other forms are neutral and put no special emphasis on that idea.
Compare the following:
He gave me a rose, he gave her a chrysanthemum.
It was a rose he gave me, he gave her a chrysanthemum.
