Fortunately for both me & the Mrs we haven't had any of the issues most people seem to be posting about, or any others for that matter, I just found the stock sms app ugly as sin.
As far as email goes I have my Gmail setup, along with two Hotmail accounts, push working on all as it should, no issues as of yet.
I would suggest try refreshing your data connection via the mobile data button on the pulldown notifications screen, see if that helps, or maybe take a second look at the email settings.