Webmail is accessing your emails through your browser and this is possible through Safari on the iPhone or iPad. You cannot sync to webmail. However since all of the mails remain on the server which ever device you use will see the emails and will seem like a sync.
If you want to use the email app on the iPhone and iPad then it is best to use imap. imap will allow the iPhone, iPad and PC to see all of the emails both incoming and sent.
If you want to use imap then set up your email with your provider. Note carefully the settings provided and input these into the iPhone and iPad. You will probably have to enter them manually. You should not need the exchange bit.
Do you use Outlook (the program)?
I'm puzzled by what you mean by all of your contacts? You can back up iPhone and iPad contacts through iCloud and/or iTunes and both devices can see the same contacts.
I have a desktop, laptop, iphone and iPad all syncing imap email, contacts and calendar.