I would say "I am packing my bags for a trip".
Student or Learner
I am packing/ packing up my bags for journey.
Which one is correct here?
You are packing your bags. Packing your bags could be said to involve packing up your things in a bag, but 'packing up' often involves dirty jeans and boxes and sticky tape - you would 'pack up' before moving house. The phrasal verb often has no direct object; a teacher at the end of an art lesson might say 'Is that the time? Only five minutes to the bell. Time to pack up. Make sure everything's put back just as you found it.'