Edit: Wait, my appoligies, what is your old phone? Is it Android? I just assumed when I posted -- sorry wasnt thinking.
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Yep you should -- Have a look under the privacy settings
(from phone's home screen) press menu > settings > privacy
Also make sure you have "background data" enabled for at least your main gmail account
(from phone's home screen) press menu > settings > accounts and sync
I'd leave it like that for a day or two to be sure it syncs OK