What Paul said...
It depends what your old email is, but most online (as in ones you use a web browser to access) allow you to export your contacts. Gmail certainly does. Similarly, most Email clients also allow you to export your contacts. It's just a matter of finding the export option. You want to export them as a CSV file.
If you need help with this, come back with more details about your account and what it is you're wanting to transfer (i.e. just email addresses, or full contact information like phone numbers and addresses).
Once you have them exported as a CSV file, you go to GMail, Contacts, Import...