Wish To Download/save All Emails From A Server?

Long story short:
Moving my website to a new host.
14 - 15 emailboxes store a lot of stuff with the current host.
Never downloaded any emails; always just signed in/read on server.

Looking for a way to get those emails downloaded, backed up, saved to local computer, whatever.
Any way at all that I can easily retain the content (messages/photos)?
Oh, the "contacts" feature was never used, and I ALSO hope to retain some type of email contact list
And, PS: I do not even know how to use a desktop email client, if that is a thing (I think it is).

My mail was squirrel mail, but it was changed recently to Roundcube. I think it can be changed back if that makes anything any easier. I am not very knowledgeable about Roundcube features, having just gotten it.

So most of the email originally came into one email box. The others were all made as holding areas when the main incoming box filled up, mainly for each year's messages which were then forwarded into them maybe 8 or 10 at a time.

Does anyone know how I can do any of this?

 

Oh and PS I no longer can access the email account used to have a profile here over a decade ago so the email address on the account is going to be updated now before I log off. So if you never hear anything from me, it means I messed that up too
But nevertheless, I WILL be here seeing if anyone offers assistance!

I also want to add that I ate a delicious meal recently in a really nice restaurant about 6 miles from the physical mailing address noted on this site. It was called Daniel's Grill.

 

You could use an email client like Postbox or Thunderbird or a program like Mailstore (https://www.mailstore.com/en/products/mailstore-home/) for IMAP or POP mail servers.

 

Thank you! The main thing I did not know was if it would only grab NEW incoming emails, or if when I set it up I can then save ALL the old emails in there to view. Your reply indicates it will do what I want. (I really would never need to USE the emails - like replying to the old ones - I just want to be able to ACCESS them for information. Again, thanks and I will take a looks at Mailstore. (Thunderbird sounds a bit confusing to me.)