Having just done this myself I would HIGHLY recommend transferring email using a IMAP based tool. Any POP based tool will not retain any of the folder organization you may have in your Gmail mailbox.

Gmail implements folders via labels, and the only way that I know of how to pass these through to the new mailbox (whether it be Gmail or not) is by using IMAP.

I used:

https://imapsync.lamiral.info/#install

This one is free if you don't need support. There are others.

The email hosting service I settled for is:

https://mxroute.com

Use them only for forwarding, so their cheapest package is OK for me.