You only hinted at what the source of your anxiety about traveling is. The way I figure it, there are three main sources: 1) Anxiety about an accident occurring on the plane. 2) Worry about getting lost, delayed, or misdirected while navigating airports, train stations, etc. 3) Worry about what'll happen with the events or people or work that's awaiting you at the other end. #2 is the main one for me, so I'll share some tips on that.

1) Make a list of your travel arrangements, along with contact numbers for each company or person who's helping you along the way. Keep this information on an electronic device where it's handy, but also print a hardcopy as a backup in case your electronics get separated from you or the battery dies.

2) Research in advance how you'll make connections at airports and major train stations. Pay special attention to immigrations and customs processes. Yes, you can generally figure this stuff out on the fly, especially if the station has signs in your language, but it's more much comforting to build reasonable knowledge in advance. I see above that there are already some good pointers to this kind of info in this thread. ^

3) Will you be taking a train, bus, or taxi to get to your final address? Look up in advance what's involved in hiring these forms of transportation.

4) Your flights are long ones, so be prepared. Be sure to pack: A) plenty of things to etnertain yourself with, such as book, e-books, and computer games; B) some food, such as fruit or nuts or energy bars, in case there's no food you like available for sale; and C) an empty water bottle you can fill up after clearing US security checkpoints.