It will depend upon the email service you are using; however, the majority of popular webmail services and email client allow for the ability to export and import contacts. Webmail services and clients, listed below, for example, allow contacts to be imported into a new email account,
If you can export/import contacts, then the process should be simple: there will be an option within the the "menu/file" tab for importing/exporting, which should then create a CSV file for you (database file). You may need to read the help files of your new email service/host/client to find the correct format for the CSV file; for example, selecting the CSV file in a "comma separated" syntax.
Once you have created a CSV file of your contacts in your previous email service/client, then it should be a straight forward procedure to input it into your new email provider/service/client. There will be a "contacts" "import/export" link in some portion of the GUI (General User Interface): which you will simple have to select, then browse and locate your contact's CSV file, and click to upload it.