First get yourself a free email account with a reliable third-party provider. (i.e. DON'T use your new ISP's email service or you'll be faced with the same problem again if you change your ISP in the future). I recommend using GMX:
https://www.gmx.co.uk
Log into your 'old' email account using your web browser on a PC. (i.e. not, say, via an Android phone app). Look for an option to export your contacts list. (Post a question here if you can't find it, telling us who your 'old' ISP is). If you're offered a choice of format, choose '.csv'. Save the exported file onto your hard drive.
Exit your 'old' account and log into your 'new' one (with GMX or whatever). Look for an option to import your contacts list. (With GMX it's via a wrench icon in 'Contacts'). Navigate to the location of the file you exported from your old account, click on it and import it.
You'll then have all of your old contacts available to you in your new account. Send an email to everyone in your contacts list, notifying them of your new address.
Then use the 'mail collector' facility in your new account to get any mail addressed to your old account automatically forwarded to your new one.
ISPs vary in their policies regarding the email accounts of departing customers. Some simply switch off the service immediately (in which case any mail sent to an old address will just be 'bounced' back to the sender) but most allow a short period of time whereby incoming mail can still be accessed (or, more importantly here, forwarded to a new account). The account will hen be closed by the old ISP. (i.e. you don't need to do anything yourself).