My family's been generally really supportive but my mom's having issues. I reminded her that I can phone cheaply (and that the phone works both directions) and she can always come visit, assuming she gets her passport, so it's not that far away. As an example of how she's dealing with things (I posted this somewhere else last week, I think), I phoned her one day and she answered the phone with, "Hi, are you calling to tell me you've met some rich doctor who's the man of your dreams and you aren't moving after all??"
It's a good thing to remember that this is your life and your adventure. Would they
really want you to stay where you are and miss this chance to learn new things and grow and discover just so that they're comfortable? My mom was laying on the guilt really thickly one day and I asked her that. Guilt works both ways, you know.
Anyway, it really gave her some perspective. She still isn't happy that I'm going, but at least she doesn't voice it so much.
I made sure to get some traveling in, like you're thinking, to spend time with all of my family before I go. I think once she can see how happy I am there, My mom'll settle in some. Hopefully the same will happen with your family. If not, there's nothing you can do about it, to be honest. Just be happy--it's contagious.