I'll be going there on an enjoyable culture trip I will be going on an enjoyable culture trip there

Not quite. "I'll be there to enjoy the culture trip." "I'm going to enjoy the culture trip there." In this sentence, you need "the" or "a" before "culture trip". It is a hard part of English; the/a. It's not something learners will really get until they are into advanced English, I have found.

I would also say you should change "culture" to "cultural," so "I'm going to enjoy a cultural trip there," but in general the sentence is worded somewhat strangely. A more native way to say it would be "I'm looking forward to the cultural trip." or "I'm looking forward to experiencing the culture there." Are you talking about a vacation?