"I will be going......" is a form of the future continuous tense ("CFT"), which according to EnglishClub.com is used to express action at a particular moment in the future. It is said that the following rules apply to the use of CFT:
- The action will start before that moment but it will not have finished at that moment.
- When we use CFT, our listener usually knows or understands what time we are talking about.
Returning to my sentence, I think FCT is appropriate to express my upcoming trip as I will continue to go from New York to Los Angeles. It would be a long and extended trip. Nonetheless, I did not follow the second rule applicable to CFT as I did not mention as to the relevant time of my trip. In the circumstances, I will revise my sentence to "I will be going to New York and Los Angeles in June". It is to tell you that I would be in the middle of my trip in June. Further you may expect that I would leave before June and would not finish the trip in June.
I am grateful to my friend for raising the question.