In all scenarios it's worth taking a moment to ask your employer what to do with these emails, depending on your location and market the company may have a legal obligation to retain your email for an extended period of time. (otherwise they could get in trouble, and you could catch the blame)
If your work email is on a personal account
If your company has it's own email you should forward any work relevant emails to your work email address. Then trash all emails that are no longer relevant. If there are any questions on what should and shouldn't be forward, it's better to send stuff they don't need than miss what they do. (again you can ask them)
Once the important stuff is all forward you can do house keeping as you see fit.
If this is a work email address (not third party)
If your company handles your email you can take the time to clean out personal stuff, but honestly I wouldn't make a huge effort of it. Likely immediately after you announced your resignation IT made a full back up of your email "just in case", perhaps as a small operation they might not have, but most companies tend to have these sort of policies, especially in areas where they have a legal obligation to retain email for a period of time.
DO NOT COPY EMAILS!!! it's okay to copy a few personal emails, but anything work related is company property that they've paid you for. While being on good terms it's unlikely to be an issue, but if things go sour it can be considered stealing company secrets. (or similar) It's best just to leave on good terms and leave anything you did behind you unless they explicitly give you permission to take something.
Not a big deal
Honestly as long as you didn't email your Social Security Number, Credit card numbers, or other such dangerous information I really wouldn't put much effort into this. If you DID send such information delete it, and don't send that kind of stuff again. Email is not a secure form of communication by default think of it like a post card, anyone watching as it goes by can see it.