1. 처음으로 is "for the first time". So the first sentence means I went abroad for the first time ("in my life" can be implied) 2. 처음에 is "first". So the second sentence means First I went aborad ( And next I did something else~~blah blah~~ kinda thing will be followed) It is used to explain something in order of time you do or it happens.

Hi 처음으로 means "for the first time" whereas 처음에 means "at first" 처음으로 해외여행을 갔다 would be, I travelled overseas "for the first time" 처음에 해외여행을 갔다 would be, "At first", I travel overseas I hope this helps, Cheers