I was in this situation a while back.
When I decided I was going to switch I opened a free gmail account. During the month's notice I had to give I changed email details on all the relevant accounts.
This meant that nothing had to be done in a day, I could make sure I had remembered every site that required action for the change and I could notify everyone who needed to know.
Gmail is web based so can be accessed from any computer in the world, and the address will stay with me, regardless of internet provider.
I still use gmail - I love it.