It's not so much transferring as syncing. Settings > email + accounts > add account. After you add the other account (like a Gmail) tap the account name and choose what to sync from the list... The Gmail (or whatever account) gets turned into an App on your App list that can be pinned to Start and you can customize how it acts in Settings > notifications + actions.
NOW if you really want to just transfer the contacts do a web search for "Import <first email> contacts to <second email>" and look at what needs to be done in a browser.
Having seen this kind of activity go very wrong quite a few times, download a "csv" backup of your Contacts from the Gmail account before doing anything else.